﻿ ﻿THE SAT MATH TEST: THE HEART OF ALGEBRA - McGraw-Hill Education SAT 2017 Edition (Mcgraw Hill's Sat) (2016)

## McGraw-Hill Education SAT 2017 Edition (Mcgraw Hill's Sat) (2016)

### THE SAT MATH TEST: THE HEART OF ALGEBRA

1. Working with Expressions
2. Working with Linear Equations
3. Working with Inequalities and Absolute Values
4. Working with Linear Systems

The SAT Math: Heart of Algebra

Why is algebra so important on the SAT Math test?

About 36% (21 out of 58) of the SAT Math questions fall under the category called the “Heart of Algebra.” Questions in this category test your ability to

analyze, fluently solve, and create linear equations, inequalities, [and] systems of equations using multiple techniques .

These questions will also assess your skill in

interpreting the interplay between graphical and algebraic representations [and] solving as a process of reasoning .

The specific topics include

• creating and solving linear equations in one and two variables
• graphing and interpreting linear equations
• creating, interpreting, and solving linear systems
• graphing and solving inequalities and systems of inequalities
• interpreting and solving algebraic word problems

Why are these skills important?

Algebra is an essential tool of quantitative analysis not only in math but also in subjects like engineering, the physical sciences, and economics. When describing the relationships between or among different quantities, or exploring the nature of unknown quantities, algebra provides essential tools for analyzing and solving problems. Most colleges consider fluency in algebra to be a vital prerequisite to a college-level liberal arts curriculum.

Sound intimidating? It”s not.

If you take the time to master the four core skills presented in these 13 lessons, you will gain the knowledge and practice you need to master even the toughest SAT Math “Heart of Algebra” questions.

Skill 1: Working with Expressions

Lesson 1: Using algebraic expressions

To solve tough SAT math problems, you must be fluent in defining, manipulating, and analyzing algebraic expressions.

Corrine drives to her office at an average speed of 50 miles per hour. When she returns home by the same route, the traffic is lighter and she averages 60 miles per hour. If her trip home is 10 minutes shorter than her trip to her office, what is the distance, in miles, from Corrine”s home to her office?

(Medium-hard ) Why does everyone hate “word problems” like this one? For most of us, the problem is that the equations aren”t “set up” for us—we have to set them up ourselves, which can be a pain in the neck. But we can make these problems much easier by breaking them down into clear steps.

Key Steps to Solving Tough Algebraic Problems

Solving tough problems in mathematics and science frequently involves four essential steps:

1. identify the relevant quantities in the situation
2. express those quantities with algebraic expressions
3. translate the facts of the problem situation into equations involving those expressions
4. analyze and solve those equations

Step 1. Identify: In this problem, there are six relevant quantities:

• the speed from home to work
• the distance from home to work
• the time it takes to get from home to work
• the speed from work to home
• the distance from work to home
• the time it takes to get from work to home

This may seem like a lot, but as we will see, keeping track of them is quite manageable.

Step 2. Express: The problem gives us enough information to express all six quantities in terms of only two “unknowns.” If d is the distance, in miles, from her home to her office, and t is the time, in hours, it takes her to get home from the office, then we can express our six quantities, respectively, as

Step 3. Translate: To translate the facts of this problem into equations, we must know the formula distance = average speed × time . Applying this to each trip gives us

Step 4. Analyze and Solve: We have now reduced the problem to a “two by two system,” that is, two equations with two unknowns. Since the number of equations equals the number of unknowns, we should be able to solve for those unknowns. (In Lessons 12 and 13, we will review these concepts and techniques.) Since the unknown d is isolated in both equations, substitution is simple:

Since t represents the time it took Corrine to return home, in hours, this means it took her 5/6 hours (or 50 minutes) to get from her office to her home, and 5/6 hour + 1/6 hour = 1 hour to get to her office from home. But remember, the question asks for the distance from her home to her office, which we can find by substituting into either of our equations:

50(5/6 + 1/6) or 60(5/6) = 50 miles

Lesson 2: The Laws of Arithmetic

When expressing or simplifying a quantity, you frequently have many options. For instance, the expression 4x 2 – 12x can also be expressed as 4x (x – 3). Similarly, 3.2 can be expressed as 16/5 or 3 or 32/10. Which way is better? It depends on what you want to do with the expression. Different forms of an expression can reveal different characteristics of that quantity or the equation in which it appears. To gain fluency in expressing quantities, you must understand the Laws of Arithmetic.

What is the value of

To simplify complex expressions, you must know the Order of Operations:

PG-ER-MD-AS

Step 1: PG (parentheses and other grouping symbols, from innermost to outermost and left to right)

Since this expression contains no parentheses, we don”t have to worry about “grouped” operations, right? Wrong! Remember that fraction bars and radicals are “grouping symbols” just like parentheses are .

In other words, we can think of this expression as

If a set of parentheses contains only one operation, then we simply do that operation:

If it contains more than one operation, then we must move on to the next step.

Step 2: ER (exponents and roots , from innermost to outermost and left to right)

Do any of the parentheses contain exponents or roots? Yes, so we must perform that operation next:

Step 3: MD (multiplication and division , from left to right)

Next, we do any multiplication inside the parentheses:

Step 4: AS (addition and subtraction , from left to right)

Now we do any addition and subtraction left in the parentheses:

Once all the “grouped” operations are completed, we run through the order of operations once again to finish up. Exponents or roots? No. Multiplication or division? Yes:

1.875 + 2

Addition or subtraction? Yes: 1.875 + 2 = 3.875

What is the sum of the first 100 positive integers?

(Hard ) Here, following the order of operations would be, shall we say, less than convenient: it would require 99 computations. Even with a calculator, it would be a pain. But here is a much simpler method:

Original expression:

1 + 2 + 3 + 4 + … + 97 + 98 + 99 + 100

Rearrange and regroup:

(1 + 100) + (2 + 99) + (3 + 98) + … + (50 + 51)

Simplify:

(101) + (101) + (101) + … + (101)

Since we have 50 pairs, this equals:

50(101)

Simplify:

5,050

This gives us exactly the same result as the order of operations would give, but with just a few simple calculations. How did we do it? By using three more laws of arithmetic: the commutative law of addition , the associative law of addition , and the distributive law of multiplication over addition .

Use the Laws of Arithmetic to simplify expressions or reveal their properties.

e.g., 3 + 8 + 17 + 12 = 3 + 17 + 12 + 8

The Commutative Law of Multiplication

When multiplying, order doesn”t matter.

e.g., 2 × 16 × 50 × 3 = 3 × 16 × 50 × 2

e.g., 1 + 100 + 2 + 99 + 3 + 98 + … + 50 + 51 = (1 + 100) + (2 + 99) + (3 + 98) + … + (50 + 51)

The Associative Law of Multiplication

When multiplying, grouping doesn”t matter.

e.g., 1 × 2 × 3 × 4 × 5 = (1 × 2 × 3) × (4 × 5)

The Distributive Law of Multiplication over Addition

When multiplying by a grouped sum, you don”t have to do the grouped sum first; you can multiply first, as long as you distribute the multiplication over the entire sum.

e.g., 5(20 + 7) = 5 × 20 + 5 × 7 = 100 + 35 = 135

Which of the following is equivalent to 3(34 × 53 )? [No calculator]

1. A) 3(34) × 3(53)
2. B) 94+ 153
3. C) 94× 153
4. D) 35× 53

Before making your choice, check the laws of arithmetic; don”t make up your own laws. Which laws of arithmetic can we use? Since the expression is a product, we can use the commutative law of multiplication and jumble up the terms, or the associative law of multiplication and regroup the terms any way we want (or not at all). Using the associative law gives us

Don”t “over-distribute.”

Were you tempted to choose (A), (B), or (C) in the question above? If so, you are not alone. You are simply the victim of one of the most common mistakes in algebra: over-distribution. It comes from a misinterpretation of the Law of Distribution. The correct law is

When multiplying by a grouped sum, you don”t have to do the grouped sum first; you can multiply first, as long as you distribute the multiplication over the entire sum .

It is not

If something is outside parentheses, just bring it inside and distribute .

Look at these examples of “over-distribution” and verify that they are incorrect:

3(2 × 5) is not equal to (3 × 2) + (3 × 5) or (3 × 2) × (3 × 5)

(2 + 3)2 is not equal to 22 + 32

If x ≠ 0, which of the following equals

1. A)
2. B)
3. C) 2x2+ 2
4. D) 4x+ 2

(Medium )

Distributive Law:

Division by a number is multiplication by its reciprocal:

Distributive Law:

So the correct answer is (D). Look at each step carefully and notice how each one uses a particular Law of Arithmetic. In particular, notice that the “combining of like terms” in steps 1–3 is really an example of commuting, associating, and (un)distributing. Even more interesting, notice that steps 4–5 show that division distributes just like multiplication does.

You can also distribute division over addition just as you can distribute multiplication.

How many distinct values of x are solutions to the equation x 2 + 4 = −4x ?

1. A) none
2. B) one
3. C) two
4. D) three

(Medium ) You might recognize that this equation is a quadratic equation (which we will discuss in much more detail in Chapter 9 ) and remember that such equations usually have two distinct solutions, but not always , so we must look at this equation more carefully.

Step 2 might seem a bit mysterious. Why did we write 4x as 2x + 2x ? Here we are using the Product-Sum Method for factoring quadratics, which is explained in a bit more detail in Chapter 9 , Lesson 4. For now, though, just notice that each step follows a particular Law of Arithmetic.

If the product of two numbers is 0, then one of those numbers must be 0. (This is the Zero Product Property .) Therefore x + 2 = 0 and so x = −2. Since the other factor is the same, we only get one solution to this equation, and the answer is (B).

To check the equation in step 5, we can FOIL the product of binomials on the left side to make sure we get the same expression we had back in step 1: (x + 2)(x + 2) = x 2 + 4x + 4, which is precisely the expression we started with in step 1, confirming that our work is correct.

This means that the factoring process in steps 2–5 can be thought of as un-FOILing. We will look at this method of factoring more carefully in Chapter 9 .

Make sure you know how to FOIL and un-FOIL.

FOILing is simply the shortcut for multiplying two binomials, which requires applying the distributive law twice. For example:

Exercise Set 1 (No Calculator)

1

(1 − (1 − (1 − 2))) − (1 − (1 − (1 − 3))) =

2

When 14 is subtracted from 6 times a number, 40 is left. What is half the number?

3

Four consecutive even numbers have a sum of 76. What is the greatest of these numbers?

4

If , then 10x + 12 =

5

What number decreased by 7 equals the opposite of five times the number?

6

If 5d + 12 = 24, then 5d − 12 =

7

If , then y + 5 =

8

The product of x and y is 36. If both x and y are integers, then what is the least possible value of xy ?

1. A) −37
2. B) −36
3. C) −35
4. D) −9

9

If a factory can manufacture b computer screens in n days at a cost of c dollars per screen, then which of the following represents the total cost, in dollars, of the computer screens that can be manufactured, at that rate, in mdays?

1. A)
2. B)
3. C)
4. D)

10

Which of the following is equivalent to 5x (2x × 3) − 5x 2 for all real values of x ?

1. A) 5x2+ 15x
2. B) 25x2
3. C) 5x2− 15x
4. D) 10x2× 15x − 5x 2

11

The symbol Ο represents one of the fundamental operators: +, −, ×, or ÷. If (x Ο y ) × (y Ο x ) = 1 for all positive values of x and y , then Ο can represent

1. A) +
2. B) ×
3. C) −
4. D) ÷

Exercise Set 1 (Calculator)

12

The difference of two numbers is 4 and their sum is 14. What is their product?

13

If x + y − 1 = 1 − (1 − x ), what is the value of y ?

14

If 3x 2 + 2x = 40, then 15x 2 + 10x =

15

Ellen is currently twice as old as Maria, but in 6 years, Maria will be 2/3 as old as Ellen. How old is Ellen now?

16

If 2x − 2y = 5 and x + y = 6, what is the value of x 2y 2 ?

17

On a typical day, a restaurant sells n grilled cheese sandwiches for p dollars each. Today, however, the manager reduced the price of grilled cheese sandwiches by 30% and as a result sold 50% more of them than usual. Which of the following represents the revenue for today”s grilled cheese sandwich sales, in dollars?

1. A) 0.5np− 0.3
2. B) 1.05np
3. C) 1.20np
4. D) 1.50np

18

For all real numbers x and y , 4x (x ) − 3xy (2x ) =

1. A) 12x2y (x − 2y )
2. B) 2x2(2 − 3y )
3. C) 2x2(2 + 3y )
4. D) 4xy(x− 3y )

19

If a = 60(99)99 + 30(99)99 , b = 99100 , and c = 90(90)99 , then which of the following expresses the correct ordering of a , b , and c ?

1. A) c<b < a
2. B) b<c < a
3. C) a<b < c
4. D) c<a < b

20

Which of the following statements must be true for all values of x , y , and z ?

1. (x + y ) + z = (z + y ) + x
2. (xy ) − z = (zy ) − x

III. (x ÷ y ) ÷ z = (z ÷ y ) ÷ x

1. A) I only
2. B) I and II only
3. C) I and III only
4. D) II and III only

21

Carlos began with twice as much money as David had. After Carlos gave \$12 to David, Carlos still had \$10 more than David. How much money did they have combined at the start?

1. A) \$34
2. B) \$68
3. C) \$102
4. D) \$108

No Calculator

1 . 1

(1 − (1 − (1 − 2))) − (1 − (1 − (1 − 3)))

Parentheses:

(1 − (1 − (−1))) − (1− (1 − (−2)))

Next parentheses:

(1 − (2)) − (1 − (3))

Next parentheses:

(−1) − (−2)

Subtract:

−1 + 2 = 1

2 . 9/2 or 4.5

6x − 14 = 40

6x = 54

Divide by 6:

x = 9

Multiply by :

3 . 22 Let n be the least of these numbers. The sum of four consecutive even numbers is therefore n + (n + 2) + (n + 4) + (n + 6) = 76.

Simplify:

4n + 12 = 76

Subtract 12:

4n = 64

Divide by 4:

n = 16

Therefore the largest of these numbers is 16 + 6 = 22.

4 . 28

Multiply by 4:

10x + 12 = 28

5 . 7/6 or 1.16 or 1.17

x − 7 = −5x

Subtract x :

−7 = −6x

Divide by –6:

6 . 0

5d + 12 = 24

Subtract 24:

5d − 12 = 0

7 . 5

Subtract y 2 :

Multiply by −5/3:

y 2 = 0

Take square root:

y = 0

y + 5 = 5

8 . C If xy = 36 and x and y are integers, then x and y are both factors of 36. In order to minimize the value of xy , we must find the greatest separation between x and y . The greatest separation between a factor pair is 1 − 36 = −35.

9 . A We should regard this as a “conversion” problem from m days into a corresponding number of dollars .

10 . B Original expression:

5x (2x × 3) − 5x 2

Parentheses:

5x (6x ) − 5x 2

Multiply:

30x 2 − 5x 2

Subtract:

25x 2

Remember: The Law of Distribution does not apply in the first step, because the grouped expression doesn”t include addition or subtraction.

11 . D The simplest approach is perhaps to choose simple values for x and y , like 2 and 3, and see which operator yields a true equation. Since (2 ÷ 3) × (3 ÷ 2) = 1, the answer is (D).

Calculator

12 . 45

ab = 4

a + b = 14

2a = 18

Divide by 2:

a = 9

Substitute a = 9:

9 + b = 14

Subtract 9:

b = 5

Evaluate ab :

ab = 9 × 5 = 45

13 . 1

x + y − 1 = 1 − (1 − x )

Distribute:

x + y − 1 = 1 − 1 + x

Subtract x :

y − 1 = 1 − 1

Simplify:

y − 1 = 0

y = 1

14 . 200

3x 2 + 2x = 40

Multiply by 5:

15x 2 + 10x = 200

15 . 12 Let e = Ellen”s current age and m = Maria”s current age.

Ellen is twice as old as Maria:

e = 2m

In 6 years, Maria will be 2/3 as old as Ellen:

Substitute e = 2m :

Multiply by 3:

3m + 18 = 2(2m + 6)

Distribute:

3m + 18 = 4m + 12

Subtract 3m and 12:

6 = m

Therefore e = 2m = 2(6) = 12.

16 . 15 First equation:

2x − 2y = 5

Divide by 2:

xy = 2.5

Second equation:

x + y = 6

Multiply:

(xy )(x + y ) = x 2y 2 = (2.5)(6) = 15

Alternately, we could solve the system using either substitution or linear combination and get x = 4.25 and y = 1.75, and evaluate x 2y 2 = (4.25)2 − (1.75)2 = 18.0625 − 3.0625 = 15.

17 . B The revenue is equal to the number of items sold times the price per item. If the restaurant typically sells n sandwiches per day, but today sold 50% more, it sold 1.5n sandwiches. If the price p was reduced 30%, today”s price is 0.70p . Therefore, the total revenue is (1.5n )(0.70p ) = 1.05np .

18 . B

4x (x ) − 3xy (2x )

Multiply:

4x 2 − 6x 2 y

Largest common factor:

2x 2 (2 − 3y )

19 . D Although a calculator is permitted for this question, most calculators will give an “overflow error” when trying to calculate numbers like 99100 , because they”re just too large. However, comparing these numbers is straightforward if we can express them in a common format.

20 . A Only statement I is true, by the Commutative and Associative Laws of Addition. Choosing simple values like x = 1, y = 2, and z = 3 will demonstrate that statements II and III do not yield true equations.

21 . C Let x = the number of dollars David had to start. If Carlos started with twice as much money as David, then Carlos started with 2x dollars. After Carlos gave David \$12, Carlos had 2x − 12 dollars and David had x + 12 dollars. If Carlos still had \$10 more than David, then

2x − 12 = 10 + x + 12

Simplify:

2x − 12 = x + 22

2x = x + 34

Subtract x :

x = 34

Therefore, David started with \$34 and Carlos started with 2(\$34) = \$68, so they had \$34 + \$68 = \$102 combined to start.

Lesson 3: Simplifying expressions and operations

If x and y are positive numbers and 3x − 2y = 7, what is the value of

1. A)
2. B)
3. C)
4. D)

(Easy ) Working with algebraic equations doesn”t always mean “solving for x .” Notice that this particular question doesn”t ask for the values of x or y , but rather for the value of a more complicated expression. This may seem harder, but it”s actually pretty simple if you understand the Law of Substitution

The Law of Substitution

If two expressions are equal, then you may substitute one for the other at any point in the problem.

How does this help us here? Notice that if we simply add 2y to both sides of the equation, we get

3x − 2y = 7

3x = 2y + 7

Therefore, by the Law of Substitution, we can substitute 3x for 2y + 7 or vice-versa. Since 2y + 7 appears in the expression we are asked to evaluate, it makes sense to replace it with 3x :

Substitute 3x for 2y + 7:

Simplify:

When a question asks you to analyze a complex expression, don”t be intimidated. Look for simple relationships that allow you to simplify them using techniques like the Law of Substitution.

Increasing a positive number x by 25% and then decreasing the result by 50% is equivalent to dividing x by what number?

1. A) 1.333
2. B) 1.5
3. C) 1.6
4. D) 1.625

(Medium ) Increasing a quantity by 25% is equivalent to multiplying it by 1.25, because the final amount is 125% of the original amount (Chapter 8 , Lesson 8). Decreasing a quantity by 50% is equivalent to multiplying it by .5, because the final amount is 50% of the original amount. Therefore, performing both changes is equivalent to multiplying by 1.25 × 0.50, or 0.625, which is equal to 5/8. But the question asks us for the equivalent division . Here, we need to remember a simple rule: multiplying by a number is equivalent to dividing by its reciprocal. Therefore, multiplying by 5/8 is the same as dividing by 8/5, which is 1.6. Therefore, the correct answer is (C).

Every operation can be expressed in terms of its inverse operation.

And here are two more handy equivalences:

If what is the value of m + n ?

(Easy ) When a problem includes a complicated expression, we should try to simplify it, but always keep an eye on what the question is asking. In this case, simplifying to find the value of m + n requires knowing some factoring identities.

Useful factoring identities

The difference of squares equals the product of conjugates:

x 2y 2 = (x + y )(xy )

Perfect square polynomials:

x 2 + 2ax + a 2 = (x + a )(x + a )

x 2 − 2ax + a 2 = (xa )(xa )

Lesson 4: Using conversion as a problem-solving tool

Niko is 27 inches shorter than his father, who is 5 feet 10 inches tall. How tall is Niko? (1 foot = 12 inches)

1. A) 3 feet 4 inches
2. B) 3 feet 6 inches
3. C) 3 feet 7 inches
4. D) 3 feet 10 inches

(Easy ) Solving this problem requires unit conversions . To convert inches to feet, we multiply by the conversion factor (1 foot/12 inches). To convert feet to inches, we multiply by its reciprocal (12 inches/1 foot). If Niko”s father is 5 feet 10 inches tall, he is 5 feet × (12 inches/1 foot) + 10 inches = 70 inches tall. If Niko is 27 inches shorter, he is 70 − 27 = 43 inches tall, which is equivalent to 43 inches × (1 foot/12 inches) = 3 7/12 feet, or 3 feet 7 inches, so the correct answer is (C).

Conversion factors as problem-solving tools

A conversion factor is simply a fraction in which the quantities in the numerator and the denominator represent equal quantities. Sometimes the equivalence is universal —for instance, 1 pound is always equal to 16 ounces—and sometimes it is problem-specific —for instance when a machine pump waters at a rate 3 gallons per hour, 1 hour of pumping is “equal” to 3 gallons being pumped.

If a factory can manufacture b computer screens in n days at a cost of c dollars per screen, then which of the following represents the total cost, in dollars, of the computer screens that can be manufactured, at that rate, in mdays?

1. A)
2. B)
3. C)
4. D)

(Medium ) This problem, from the previous exercise set, can be solved in several different ways. One method is to simply choose values for the unknowns and turn the problem into an arithmetic problem instead of an algebra problem. But here we will look at it as a conversion problem.

We can think of this problem as being a “conversion” from a quantity of days to an equivalent quantity of dollars . We are given that this factory is working for m days, so we write this quantity down, including the units, and we multiply by the conversion factors until we get dollars:

So the correct answer is (A).

Exercise Set 2 (No Calculator)

1

If bag A weighs 4 pounds 5 ounces and bag B weighs 6 pounds 2 ounces, how much heavier, in ounces , is bag B than bag A? (1 pound = 16 ounces)

2

If , what is the value of ?

3

If x − 2y = 10 and x ≠ 0, what is the value of

4

If ab = 4 and a 2b 2 = 3, what is the value of a + b ?

5

If 6 gricks are equivalent to 5 merts, then 2 merts are equivalent to how many gricks?

6

If the function {x } is defined by the equation {x } = (1 − x )2 , what is the value of {{4}}?

7

If and , what is the value of

8

(x − 9)(xa ) = x 2 − 4ax + b

In the equation above, a and b are constants. If the equation is true for all values of x , what is the value of b ?

1. A) −27
2. B) −12
3. C) 12
4. D) 27

9

If , what is the value of x ?

1. A)
2. B) −7
3. C)
4. D)

10

(p + 2)2 = (p − 5)2

The equation above is true for which of the following values of p ?

1. A) −2 and 5
2. B) 2 and −5
3. C) 1.5 only
4. D) 5 only

11

If for all positive values of m and n , then which of the following is equal to x ?

1. A)
2. B)
3. C)
4. D)

Exercise Set 2 (Calculator)

12

Let m be a positive real number. Increasing m by 60% then decreasing the result by 50% is equivalent to dividing m by what number?

13

What is the sum of the first 50 positive even integers?

14

Three years ago, Nora was half as old as Mary is now. If Mary is four years older than Nora, how old is Mary now?

15

If 2/3 of the seats at a football stadium were filled at the beginning of the game, and at halftime 1,000 spectators left, leaving 3/7 of the seats filled, what is the total number of seats in the stadium?

16

If three candy bars and two gumdrops cost \$2.20, and four candy bars and two gumdrops cost \$2.80, what is the cost, in dollars, of one gumdrop?

17

If , what is the value of x −1?

18

Subtracting 3 from a number and then multiplying this result by 4 is equivalent to multiplying the original number by 4 and then subtracting what number?

19

In a poker game, a blue chip is worth 2 dollars more than a red chip, and a red chip is worth 2 dollars more than a green chip. If 5 green chips are worth m dollars, then which of the following represents the value, in dollars, of 10 blue chips and 5 red chips?

1. A) 50 + 3m
2. B) 18 + 60m
3. C) 40 + 3m
4. D) 28 + 20m

20

A train travels at an average speed of 50 miles per hour for the first 100 miles of a 200-mile trip, and at an average of 75 miles per hour for final 100 miles. What is the train”s average speed for the entire trip?

1. A) 58.5 mph
2. B) 60.0 mph
3. C) 62.5 mph
4. D) 63.5 mph

21

Which of the following is equivalent to 3m (m 2 × 2m ) for all real values of m ?

1. A) 3m2+ 6m
2. B) 3m2× 6m
3. C) 3m3× 6m 2
4. D) 6m4

22

If the cost of living in a certain city increased by 20% in the 10 years from 1980 to 1990, and increased by 50% in the 20 years from 1980 to 2000, what was the percent increase in the cost of living from 1990 to 2000?

1. A) 15%
2. B) 20%
3. C) 25%
4. D) 30%

No Calculator

1 . 29 4 pounds 5 ounces = 4(16) + 5 = 69 ounces, and 6 pounds 2 ounces = 6(16) + 2 = 98 ounces. Therefore, bag B weighs 98 − 69 = 29 ounces more.

2 . 2/15 or .133

Distribute division:

Simplify:

Subtract 1:

Divide by 3:

3 . 4 Expression to be evaluated:

Given equation:

x − 2y = 10

x = 2y + 10

Substitute x = 2y + 10:

Simplify:

Factor and simplify:

4 . ¾ or .75

a 2b 2 = 3

Factor:

(ab )(a + b ) = 3

Substitute ab = 4:

4(a + b ) = 3

Divide by 4:

5 . 12/5 or 2.4

6 . 64

{4} = (1 − 4)2 = (−3)2 = 9

{{4}} = (1 − {4})2 = (1 − 9)2 = (−8)2 = 64

7 . 2 Given equation:

Distribute division:

Subtract 1:

Reciprocate:

Given equation:

Distribute division:

Subtract 1:

Multiply:

8 . D Given:

(x − 9)(xa ) = x 2 − 4ax + b

FOIL:

x 2ax − 9x + 9a = x 2 − 4ax + b

Simplify:

x 2 − (a + 9)x + 9a = x 2 − 4ax + b

If this equation is true for all x , then the coefficients of corresponding terms must be equal, so

a + 9 = 4a

Subtract a :

9 = 3a

Divide by 3:

3 = a

Therefore b = 9a = 9(3) = 27.

9 . A Given equation:

Multiply by 5x :

25 + 7x = 5x

Subtract 7x :

25 = −2x

Divide by −2:

10 . C Given equation:

(p + 2)2 = (p − 5)2

FOIL:

p 2 + 4p + 4 = p 2 − 10p + 25

Subtract p 2 :

4p + 4 = −10p + 25

14p + 4 = 25

Subtract 4:

14p = 21

Divide by 14:

p = 1.5

11 . D Given equation:

Multiply by mnx :

3x = 2(mnx )

Distribute:

3x = 2m − 2nx

3x + 2nx = 2m

Factor out x :

x (3 + 2n ) = 2m

Divide by 3 + 2n :

Calculator

12 . 1.25 Increasing a number by 60% is equivalent to multiplying it by 1.60, and decreasing a number by 50% is equivalent to multiplying it by 0.50. Therefore, performing both changes in succession is equivalent to multiplying by 1.60 × 0.50 = 0.80. Multiplying by 0.80 is equivalent to dividing by its reciprocal: 1/(0.80) = 1.25.

13 . 2,550 The sum of the first 50 positive even integers is 2 + 4 + 6 + 8 + … + 100. As with the example is Lesson 2, these numbers can be regrouped into 25 pairs of numbers each of which has a sum of 2 + 100 = 102. Therefore, their sum is 25(102) = 2,550.

14 . 14 Let n = Nora”s age now, and m = Mary”s age now. If 3 years ago, Nora was half as old

as Mary is now:

If Mary is 4 years older than Nora:

m = 4 + n

Subtract 4:

m − 4 = n

Substitute n = m − 4:

Simplify:

Multiply by 2:

2m − 14 = m

m = 14

15 . 4,200 Let x = the total number of seats in the stadium.

Subtract x :

Combine like terms:

Multiply by :

16 . 0.20 Let g = the cost, in dollars, of one gumdrop, and c = the cost, in dollars, of one candy bar.

4c + 2g = 2.80

3c + 2g = 2.20

Subtract:

c = 0.60

Substitute c = 0.60:

4(0.60) + 2g = 2.80

Simplify:

2.40 + 2g = 2.80

Subtract 2.40:

2g = 0.40

Divide by 2:

g = 0.20

17 . 6

Factor:

Multiply by −1:

Simplify:

Multiply by 2:

x − 1 = 6

18 . 12 We can just choose a number to work with, like 10. If we subtract 3 from this number then multiply the result by 4, we get 4(10 − 3) = 28. If we multiply it by 4 and then subtract a mystery number, we get 4(10) − x = 40 − x.

28 = 40 − x

Subtract 40:

−12 = −x

Multiply by −1:

12 = x

19 . A If 5 green chips are worth m dollars, then each green chip is worth m /5 dollars. If a red chip is worth 2 dollars more than a green chip, then each red chip is worth m /5 + 2 dollars. If each blue chip is worth 2 dollars more than a red chip, then each blue chip is worth m /5 + 4 dollars. Therefore, 10 blue chips and 5 red chips are worth 10(m /5 + 4) + 5(m /5 + 2) = 2m + 40 + m + 10 = 3m + 50 dollars.

20 . B The average speed is equal to the total distance divided by the total time. The total distance is 200 miles. The time for the first hundred miles is (100 miles/50 mph) = 2 hours, and the time for the second hundred miles is (100 miles/75 mph) = 4/3 hours. Therefore the total time of the trip is 2 + 4/3 = 10/3 hours, and the average speed is

21 . D

3m (m 2 × 2m )

Parentheses:

3m (2m 3 )

Multiply:

6m 4

22 . C Assume the cost of living in 1980 was \$100. If this increased by 20% from 1980 to 1990, then the cost of living in 1990 was 1.20(\$100) = \$120. If the increase from 1980 to 2000 was 50%, then the cost of living in 2000 was 1.50(\$100) = \$150. The percent increase from 1990 to 2000 is therefore

Skill 2: Working with Linear Equations

Lesson 5: Constructing and interpreting linear equations

The Horizon Resort charges \$150 per night for a single room, and a one-time valet parking fee of \$35. There is a 6.5% state tax on the room charges, but no tax on the valet parking fee. Which of the following equations represents the total charges in dollars, c , for a single room, valet parking, and taxes, for a stay of n nights at The Horizon Resort?

1. A) c= (150 + 0.065n) + 35
2. B) c= 1.065(150n) + 35
3. C) c= 1.065(150n+ 35)
4. D) c= 1.065(150 + 35)n

(Medium ) This question asks us explicitly to set up an equation to express a mathematical relationship in a word problem. Usually, this is just the first step in analyzing the situation more deeply, for instance, finding particular values of the variables that satisfy certain conditions, or interpreting the meanings of terms or coefficients in the equation, but this problem only asks us to set up the equation.

When translating verbal information into an equation, it”s helpful to take small steps. First, since the room charge is \$150 per night, the charge for n nights is \$150n . If a 6.5% tax is added to this, the room charge becomes 150n + 0.065(150n ) = 1.065(150n ). The \$35 valet parking charge is added separately, and not taxed, so the total charges are 1.065(150n ) + 35, and the correct answer is (B). Notice that this equation shows a linear relationship between cand n .

When setting up equations from word problems, try to classify the relationship (that is, linear, quadratic, exponential) between the variables , so that you can check that the equation is of the correct form. In this lesson, we will focus only on linear relationships , that is, relationships that can be expressed in the form y = mx + b .

Which of the following represents the equation of the line with an x -intercept of 6 that passes through the point (4, 4)?

1. A)
2. B) y= 2x− 4
3. C) y= −2x+ 12
4. D) y= −2x+ 6

(Easy ) This question asks you to construct the equation of a line given some facts about its graph. Start by drawing a graph (on the xy -plane) of the given information in the space next to the question. It also helps to know something about the different forms of linear equations and what they reveal about the graph of the line.

Graph of a line in the xy -plane

Forms of linear equations

Slope-intercept form: y = mx + b

Standard form:  ax + by = c

Point-slope form:yy 1 = m (xx 1 )

Features: slope = m , y -intercept = b

Features: slope = −a /b , y -intercept = c /b , x -intercept = c /a

Features: slope = m , point on line = (x 1 , y 1 )

In this problem, we are given two points on the line: (4, 4) and the x -intercept (6, 0). We can calculate the slope using the slope formula above: slope = (4 − 0)/(4 − 6) = (4)/(−2) = −2. If we use this slope and the point (6, 0), we can set up the equation in point-slope form:

Point-slope form of equation:

y − 0 = −2(x − 6)

Simplify and distribute:

y = −2x + 12

This is the equation in (C). Notice that this equation is in slope-intercept form , and reveals that this line also has a y-intercept of 12. Check this fact against your diagram, and also check that both given points, (4, 4) and (6, 0), satisfy this equation.

Lesson 6: Solving equations with the Laws of Equality

If , what is the value of x ?

1. A)
2. B)
3. C)
4. D)

(Medium ) At first glance, this doesn”t look like a linear equation. But one simple move reveals that it is:

Multiply both sides by 3x :

Distribute and simplify:

3 + 2 = 12x

Divide by 12:

5/12 = x

As this shows, sometimes solving equations requires a clever use of the Laws of Equality.

The Laws of Equality

Every equation is a balanced scale, and the Laws of Equality are simply the rules for “keeping the scale balanced,” that is, deducing other true equations. In a nutshell, the Laws of Equality say that

1. You may make changes to any equation, as long as you follow rules 2 and 3.
2. Whatever you do to one side of the equation, you must do to the other.
3. You may not perform undefined operations (like dividing by 0), or operations that have more than one possible result (like taking a square root).

If x 2 = y 2 , then which of the following must be true?

1. x = y

III. x = |y |

1. A) none
2. B) I only
3. C) I and II only
4. D) I, II and III

(Medium-hard ) This question tests your skills of deductive logic . Notice it is not asking which statements can be true, but rather which must be true . It seems that if we “unsquare” both sides of the original equation, we get the equation in I. If we divide the original equation by x on both sides, we get the equation in II. Does this mean that statements I and II are necessarily true? No, because we violated rule 3 in both cases. If x 2 = y 2 , it does not follow that x = y . Notice that x could be 2 and y could be −2. These values certainly satisfy the original equation, but they do not satisfy the equations in I or III. They do, however, satisfy the equation in II, because 2 = (−2)2 /2. However, statement II is still not necessarily true. What if x and y were both 0? This would satisfy the original equation, but 0 ≠ (0)2 /(0) because 0/0 is undefined . Therefore, the correct answer is (A).

This example teaches us two lessons:

1. Before taking the square root of both sides of an equation, remember that every positive number hastwo square roots. For instance the square root of 9 is 3 or −3.
2. Before dividing both sides of an equation by an unknown, make sure it can”t equal 0.

Lesson 7: Making and analyzing graphs of linear equations

If m is a constant greater than 1, which of the following could be the graph in the xy -plane of x + my + m = 0?

A)

B)

C)

D)

(Medium-hard ) First, we should try to get the equation into a more useful form. Let”s try the slope-intercept (y = mx + b ) form:

x + my + m = 0

Subtract x and m :

my = −xm

Divide by m :

This shows that the line has a slope of −1/m and a y -intercept of −1. Since the problem tells us that m is greater than 1, we know that the slope (−1/m ) must be between −1 and 0. The only graph that satisfies these conditions is (B).

It”s helpful to think of slope as the amount a line goes up (or down) for each step it takes to the right. Lines with a positive slope slant upward to the right, lines with a negative slope slant downward to the right, and lines with a 0 slope are horizontal . For instance, a line with slope −3 moves down 3 units for every unit step to the right.

Parallel and perpendicular lines

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are opposite reciprocals of each other.That is, if one line has a slope of a/b , its perpendicular has a slope of −b/a .

The points A (10, 4) and B (−2, k ) are 13 units apart. Which of the following equations could describe the line that contains points A and B ?

1. A) 13x+ 12y= 178
2. B) 5x+ 12y= 98
3. C) 5x− 12y= 98
4. D) 5x− 13y= −2

(Hard ) Drawing a diagram will help us analyze this problem. Although we don”t know precisely where point B is, we know it is somewhere on the line x = −2. This gives us the following picture:

Next, notice that all of the equations given in the choices are in “standard” form, and in standard form the slope of the line is −a /b . Therefore, the slopes of these lines are, respectively, (A) −13/12, (B) −5/12, (C) 5/12, and (D) 5/13. Therefore, finding the slope of the line should help us choose the correct equation. Looking at the diagram more closely, notice that it includes two right triangles, and we can find the missing side of each one using the Pythagorean Theorem, or just by noticing that they are both 5-12-13 right triangles (52 + 122 = 132 ). Putting this information into the diagram shows us that B can therefore be at (−2, 9) or (−2, −1).

Therefore, the slope (rise/run) of the line containing B 1 is −5/12, and the slope of the line containing B 2 is 5/12. This means that our answer is either (B) or (C). How do we choose between them? Just remember that the line must contain the point (10, 4). If you plug x = 10 and y = 4 into these equations, only (B) works: 5(10) + 4(12) = 98.

Always check that your solutions satisfy your equations by plugging them back into the equations to verify.

Exercise Set 3 (No Calculator)

1

If x − 2(1 − x ) = 5, what is the value of x ?

2

If f (x ) = −2x + 8, and f (k ) = −10, what is the value of k ?

3

What is the slope of the line that contains the points (−2, 3) and (4, 5)?

4

What is the slope of the line described by the equation

5

Line l is perpendicular to the line described by the equation 5x + 11y = 16. What is the slope of line l ?

6

If , what is the value of x ?

7

What is the y -intercept of the line containing the points (3, 7) and (6, 3)?

8

In the xy -plane, the graph of y = h (x ) is a line with slope −2. If h (3) = 1 and h (b ) = −9, what is the value of b ?

9

If a train maintains a constant speed of 60 miles per hour, it can travel 4 miles per gallon of diesel fuel. If this train begins a trip with a full 200 gallon tank of diesel fuel, and maintains a speed of 60 miles per hour, which of the following equations represents the number of gallons, g , left in the tank t hours into the trip?

1. A)
2. B)
3. C) g= 200 −15t
4. D)

10

The points A (2, 3) and B (m , 11), are 10 units apart. Which of the following equations could describe the line that contains points A and B ?

1. A) 8x+ 6y= 11
2. B) 8x− 6y= −2
3. C) 6x+ 8y= 36
4. D) 6x− 8y= −12

11

The figure above shows a right triangle with vertices at the origin, (5, 6) and (k , 0). What is the value of k ?

1. A)
2. B)
3. C)
4. D)

Exercise Set 3 (Calculator)

12

If the points (2, 4), (5, k ), and (8, 20) are on the same line, what is the value of k ?

13

Line l has a slope of 3 and a y -intercept of −4. What is its x -intercept?

14

If f (−1) = 1 and f (3) = 2 and f is a linear function, what is the slope of the graph y = f (x )?

15

If f (−1) = 1 and f (3) = 2 and f is a linear function, what is f (5)?

16

In the xy -plane, the graph of line n has an x -intercept of 2b and an y -intercept of −8b , where b ≠ 0. What is the slope of line n ?

17

If , what is the value of x ?

18

If the line 3x − 2y = 12 is graphed in the xy -plane, what is its x -intercept?

19

If the graphs of the equations 5x − 2y = 5 and 6x + ky = 9 are perpendicular, what is the value of k ?

20

The net profit for the sales of a product is equal to the total revenue from the sales of that product minus the total cost for the sales of that product. If a particular model of calculator sells for \$98, and the cost for making and selling n of these calculators is \$(35n + 120,000), which of the following equations expresses the net profit in dollars, P , for making and selling n of these calculators?

1. A) P= 63n− 120,000
2. B) P= 63n+ 120,000
3. C) P= 63(n− 120,000)
4. D) P= 63(n+ 120,000)

21

Which of the following represents the equation of the line with an x -intercept of 5 and a y -intercept of 6?

1. A)
2. B)
3. C)
4. D)

22

The table above shows several ordered pairs corresponding to the linear function f . What is the value of a + b ?

1. A) 12
2. B) 16
3. C) 20
4. D) It cannot be determined from the information given.

Part 1: No Calculator

1 . 7/3 or 2.33

x − 2(1 − x ) = 5

Distribute:

x − 2 + 2x = 5

Simplify:

3x − 2 = 5

3x = 7

Divide by 3:

x = 7/3

2 . 9

f (k ) = −2k + 8 = −10

Subtract 8:

−2k = −18

Divide by −2:

k = 9

3 . 1/3 or .333

4 . 10/3 or 3.33

Multiply by 2xy :

Simplify:

2y + y = 10x

Simplify:

3y = 10x

Divide by 3:

5 . 11/5 or 2.2 The slope of the given line is −5/11, so the slope of the line perpendicular to it is 11/5.

6 . 9/5 or 1.8

Multiply by 10:

(x + 1) + 4x = 10

Simplify:

5x + 1 = 10

Subtract 1:

5x = 9

Divide by 5:

x = 9/5

7 . 11 There are a variety of ways of solving this problem, but perhaps the simplest is to draw a quick sketch:

Notice that to get from (6, 3) to (3, 7) we must go left 3 units and up 4 units (in other words, the slope is −4/3). If we simply repeat this from (3, 7), we arrive at the y -intercept, which is (0, 11).

8 . 8 This line has a slope of −2 and contains the points (3, 1) and (b , −9). Therefore

Simplify:

Multiply by b − 3:

−2b + 6 = −10

Subtract 6:

−2b = −16

Divide by −2:

b = 8

9 . C Since the tanks starts with 200 gallons, the amount it has left is 200 − the number of gallons used. The number of gallons used is

10 . B Once again, a quick sketch can be very helpful. Notice that traveling from point

A (2, 3) to point B (m , 11) requires going up 8 units and right (or left) some unknown distance b . We can find b with the

Pythagorean Theorem:

82 + b 2 = 102

Simplify:

64 + b 2 = 100

Subtract 64:

b 2 = 36

Take the square root:

b = 6

Therefore, m is either 2 − 6 = −4 or 2 + 6 = 8, and the slope of this line is either 8/6 = 4/3 or 8/(−6) = −4/3. The only equation among the choices that is satisfied by the ordered pair (2, 3) and has a slope of either 4/3 or −4/3 is (B).

11 . D Recall that the slopes of perpendicular lines are opposite reciprocals. The slope of the segment from (0, 0) to (5, 6) is 6/5, so the slope of its perpendicular is −5/6.

Therefore

Cross-multiply:

−36 = 5(5 − k )

Distribute:

−36 = 25 − 5k

Subtract 25:

−61 = −5k

Divide by −5:

61/5 = k

Part 2: Calculator

12 . 12 The slope of this line is ,

therefore,

Cross-multiply:

3k − 12 = 24

3k = 36

Divide by 3:

k = 12

13 . 4/3 or 1.33 Since the slope and y -intercept are given, it is easy to express the linear equation in slope-intercept form: y = 3x − 4.

The x -intercept is the value of x on the line for which y = 0:

0 = 3x − 4

4 = 3x

Divide by 3:

4/3 = x

14 . ¼ or .25 The line contains the points (−1, 1) and (3, 2), so its slope is

15 . 5/2 or 2.5 Although we could solve this problem by deriving the linear equation, it is perhaps easier to take advantage of the result from question 14. The slope of 1/4 means that the y -coordinate of any point on the line increases by 1/3 each time the x -coordinate increases by 1. Since the x -coordinate increases by 2 between f (3) and f (5), the y -coordinate must therefore increase by 2(1/4) = 1/2, so f (5) = 2 + ½ = 2.5.

16 . 4 The line contains the points (2b , 0) and (0, −8b ); therefore, it has a slope of .

17 . 3/5 or .6

Multiply by 5x:

10 + 2 = 20x

Simplify:

12 = 20x

Divide by 20:

x = 12/20 = 3/5

18 . 4 The x -intercept is the value of x for which y = 0:

3x − 2(0) = 12

Simplify:

3x = 12

Divide by 3:

x = 4

19 . 15 Recall that the slope of a line in standard form ax + by = c is −a/b . Therefore, the slope of 5x − 2y = 5 is 5/2 and the slope of 6x + ky = 9 is −6/k . If these lines are perpendicular, then their slopes are opposite reciprocals:

Multiply by 6:

k = 30/2 = 15

20 . A The total revenue for selling n calculators at \$98 each is \$98n the cost for making and selling n calculators is \$(35n + 120,000). Therefore the profit is \$(98n − 35n − 120,000) = 63n − 120,000 dollars.

21 . C This line contains the points (5, 0) and (0, 6) and therefore has a slope of .

Since its y -intercept is 6, its slope-intercept form is or, subtracting 6 from both sides,

22 . B Since f is a linear function, it has a slope that we can call m . Recall that it”s often useful to think of the slope of a line as the “unit change,” that is, the amount that y changes each time x increases by 1. Since the x values increase by 1 with each step in our table, the y values must therefore increase by m with each step. This means that a = 8 − m and b = 8 + m . Therefore, a + b = 8 − m + 8 + m = 16.

Skill 3: Working with Inequalities and Absolute Values

Lesson 8: Understanding inequalities and absolute values

On the real number line, a number, b , is more than twice as far from −3 as it is from 3. Which of the following equations can be solved to find all possible values of b ?

1. A) |b− 3| > 2|b+ 3|
2. B) |b+ 3| > 2|b− 3|
3. C) 2|b− 3| > |b+ 3|
4. D) 2|b+ 3| > |b− 3|

Distance and absolute value

The absolute value of a number x , written as |x |, means its distance from 0 on the number line. In fact, we can use absolute value to represent the distance between any two numbers.

|xa | means the distance between x and a on the number line.

Notice that this works no matter which number is greater. For instance, the distance between 2 and 7 is |2 − 7| = |−5| = 5, which is the same as the distance between 7 and 2: |7 − 2| = |5| = 5.

Notice that an expression like |x + a | is equivalent to |x − (−a )|, which means that |x + a | can be translated as the distance between x and −a .

(Medium-hard ) We can use this definition to translate the problem. The key is to translate the statement “b is more than twice as far from −3 as it is from 3” into a statement about distances : “The distance between b and −3 is more than twice the distance between b and 3.” Notice how easily this translates into an inequality:

(The distance between b and3 ) is more than (twice the distance between b and 3 )

|b − (−3)| > 2|b − 3|

|b + 3| > 2|b − 3|

which is choice (B).

Lesson 9: Solving inequalities with the Laws of Inequality

If , what is one possible value of x ?

(Easy) This kind of inequality is called a “sandwich inequality” because the expression in the middle is between the other two, like meat between slices of bread. Working with inequalities like this one requires understanding the Laws of In equality .

The Laws of Inequality

Every inequality is a “tipped” scale, and the Laws of Inequality are simply the rules for “keeping the scale tipped the right way,” that is, deducing other true inequalities that follow from the original one. In a nutshell, the Laws of Inequality say that

1. You may make changes to any inequality, as long as you follow rules 2, 3 and 4.
2. Whatever you do to one side of the inequality, you must do to the other.
3. You may not perform undefined operations (like dividing by 0), or operations that have more than one possible result (like taking a square root).
4. If you multiply or divide both sides by a negative number, you must “switch” the direction of the inequality. This is because multiplying or dividing by a negative number involves areflection over the origin on the number line, and this reflection requires the “switch”:

So we can solve the sandwich inequality by applying the correct laws of inequality:

Multiply by −6 (the common denominator) and “switch:”

3 > 12x − 6 > 2

9 > 12x > 8

Divide by 12:

0.75 > x > 0.66 …

Therefore, any value greater than 0.666 but less than 0.750 is correct.

Which of the following must be true if

1. A) a≤ −3b
2. B) a≥ −3b
3. C) a≤ −3b< 0 or a ≥ −3b > 0
4. D) a≤ −3b< 0 or a ≥ −3b > 0

(Hard ) We might be tempted to multiply both sides of the inequality by b and get the inequality in (A). But this would be incorrect because it would ignore rule 4. We need to consider the possibility that b might be negative . Let”s think about possible solutions to the original inequality. Notice that a = 10 and b = −2 gives a possible solution, because 10/(−2) = −5 ≤ −3. But this would not satisfy the inequality in (A): 10 is not less than or equal to (−3)(−2) = 6.

To solve this inequality, we will need to consider two distinct possible conditions:

Condition 1: If b > 0, then a ≤ −3b and therefore a ≤ −3b < 0

Condition 2: If b < 0, then a ≥ −3b and therefore a ≥ −3b > 0

which is the answer in choice (D).

Lesson 10: Graphing inequalities

On the real number line, a number, b , is more than twice as far from −3 as it is from 3. Which of the following graphs represents all possible values of b ?

A)

B)

C)

D)

(Medium) We saw this scenario in Lesson 8, but now we are asked to graph the solution. Recall from Lesson 8 that this relationship is expressed by the inequality |b + 3| > 2|b − 3|. How do we translate this into a graph? The simplest way to start is to visualize the number line, and to think about a related, but simpler, question: What if b is exactly twice as far from3 as it is from 3 ? A little guessing and checking should reveal that two points work:

Notice that 9 works because 3 is the midpoint between −3 and 9, and 1 works because it is 2/3 of the way from −3 to 3. Also, you can confirm that both numbers satisfy the equation |b + 3| = 2|b − 3|. These two points now divide the line into three parts: everything less than 1, everything between 1 and 9, and everything greater than 9. A little bit of checking (just pick a number from each portion and plug it into our inequality) confirms that only the numbers in the middle portion satisfy our inequality, so the correct graph is the one in choice (D).

When graphing inequalities, it often helps to start with the graph of the corresponding equation and work from there. The graph of the equation usually provides the boundaries for the graph of the inequality.

Exercise Set 4 (No Calculator)

1

What positive number is twice as far from 10 as it is from 1?

2

If the points (2, a ) and (14, b ) are 20 units apart, what is |ab |?

3

What is the least integer n for which

4

If |x + 4| = |x − 5|, what is the value of x ?

5

What is the greatest integer value of n such that

6

What is the only integer b for which and 3b ≥ 7.5?

7

If (b + 2)2 = (b − 5)2 , what is the value of b ?

8

Which of the following statements is equivalent to the statement −4 < 2x ≤ 2?

1. A) x> −2 andx ≤ 1
2. B) x< −2 orx ≥ 1
3. C) x≥ −2 andx < 1
4. D) x≤ −2 orx > 1

9

The annual profit from the sales of an item is equal to the annual revenue minus the annual cost for that item. The revenue from that item is equal to the number of units sold times the price per unit. If n units of a portable heart monitor were sold in 2012 at a price of \$65 each, and the annual cost to produce n units was \$(20,000 + 10n ), then which of the following statements indicates that the total profit for this heart monitor in 2012 was greater than \$500,000?

1. A) 500,000 < 55n− 20,000
2. B) 500,000 > 55n− 20,000
3. C) 500,000 < 55n+ 20,000n
4. D) 500,000 < 75n− 20,000n

10

Colin can read a maximum of 25 pages an hour. If he has been reading a 250 page book for h hours, where h < 10, and has p pages left to read, which of the following expresses the relationship between p and h ?

1. A)
2. B)
3. C) 250 −p≤ 25h
4. D) 250 + 25hp

11

On the real number line, a number, x , is more than 4 times as far from 10 as it is from 40. Which of the following statements describes all possible values of x ?

1. A) x< 34 orx > 50
2. B) x> 40
3. C) 34 <x< 50
4. D) 32.5 <x< 160

Exercise Set 4 (Calculator)

12

If a < 0 and |a − 5| = 7, what is |a |?

13

If n is a positive integer and 16 > |6 − 3n | > 19, what is the value of n ?

14

What is the only integer n such that 20 − 2n > 5 and

15

What is the smallest number that is as far from 9.25 as 3 is from −1.5?

16

If |2x + 1| = 2|kx |, for all values of x , what is the value of |k |?

17

Which of the following is equivalent to the statement |x − 2| < 1?

1. A) x< 3
2. B) x< −1
3. C) 1 <x< 3
4. D) −1 <x< 3

18

If the average (arithmetic mean) of a and b is greater than the average (arithmetic mean) of c and 2b , which of the following must be true?

1. A) b> 0
2. B) a>b
3. C) a>b + c
4. D) a+c > b

19

Of the statements below, which is equivalent to the statement “The distance from x to 1 is greater than the distance from x to 3?

1. A) 1 <x< 3
2. B) x> 2
3. C) x< 2
4. D) x− 1 > 3

20

Which of the following is equivalent to the statement 4x 2 ≥ 9?

1. A) 2x> 3
2. B) x≥ 1.5 orx ≤ −1.5
3. C) |x| > 2
4. D) −1.5 ≤x≤ 1.5

21

The graph above indicates the complete solution set to which of the following statements?

1. A) |x− 3| > 3
2. B) |x| < 6
3. C) |x− 6| < 6
4. D) |x− 3| < 3

22

Which of the following is true for all real values of x ?

1. A) |x| > 0
2. B) x< 2 orx > 1
3. C) x> −2 orx < −3
4. D) x2− 1 > 0

No Calculator

1 . 4 It is helpful to plot these values on the number line and think:

The distance between 1 and 10 is 9, so clearly the number that is 9 more units to the left of 1, namely −8, is twice as far from 10 as it is from 1. However, this is a negative number so it can”t be our answer. There is one other number that is twice as far from 10 as it is from 1: the number that is 1/3 the distance from 1 to 10. This number is 4, which is 3 units from 1 and 6 units from 10.

2 . 16 From the Distance Formula,

(2 − 14)2 + (ab )2 = 202

Simplify:

144 + (ab )2 = 400

Subtract 144:

(ab )2 = 256

Square root:

|ab | = 16

3 . 8

Since n must be positive for this statement to be true, we can multiply by 9n without having to “swap” the inequality symbols:

0 < 36 < 5n

Divide by 5:

0 < 7.2 < n

Therefore, the smallest integer value of n is 8.

4 . ½ or .5 Two numbers, a and b , have the same absolute value only if they are equal or opposites. Clearly x + 4 and x − 5 cannot be equal, since x − 5 is 9 less than x + 4. Therefore they must be opposites.

x + 4 = −(x − 5)

Distribute:

x + 4 = −x + 5

2x + 4 = 5

Subtract 4:

2x = 1

Divide by 2:

x = 1/2

5 . 10

Multiply by −42 and “swap:”

2n < 21

Divide by 2:

n < 10.5

Therefore, the greatest possible integer value of n is 10.

6 . 3

3b ≥ 7.5

Divide by 3:

b ≥ 2.5

Since b is greater than or equal to 2.5, it is positive, so we can multiply both sides by 11b without “swapping” the inequality:

11 > 3b

Divide by 3:

3.67 > b

The only integer between 2.5 and 3.67 is 3.

7 . 3/2 or 1.5

(b + 2)2 = (b − 5)2

FOIL:

b 2 + 4b + 4 = b 2 − 10b + 25

Subtract b 2 :

4b + 4 = −10b + 25

14b + 4 = 25

Subtract 4:

14b = 21

Divide by 14:

b = 1.5

8 . A

−4 < 2x ≤ 2

Divide by 2:

−2 < x ≤ 1

which is equivalent to −2 < x and x ≤ 1.

9 . A The profit is the revenue minus the cost: 65n − (20,000 + 10n ) = 55n − 20,000.

10 . C If Colin can read a maximum of 25 pages an hour, then in h hours he can read a maximum of 25h pages. If he has p pages left in a 250-page book, he has read 250 − p pages. Since it has taken him h hours to read these 250 − p pages, 250 − p ≤ 25h .

11 . C

|x − 10| > 4|x − 40|

It helps to sketch the number line and divide is into three sections: the numbers less than 10, the numbers between 10 and 40, and the numbers greater than 40.

CASE 1: x < 10. It should be clear that all numbers less than 10 are closer to 10 than they are to 40, so this set contains no solutions.

CASE 2: 10 < x ≤ 40. If x is between 10 and 40, x − 10 is positive and x − 40 is negative, so |x − 10| = x − 10 and |x − 40| = −(x − 40).

|x − 10| > 4|x − 40|

Substitute:

x − 10 > −4(x − 40)

Distribute:

x − 10 > −4x + 160

5x − 10 > 160

5x > 170

Divide by 5:

x > 34

So this gives us 34 < x ≤ 40.

CASE 3: x > 40. If x is greater than 40, then both x − 10 and x − 40 are positive, so |x − 10| = x − 10 and |x − 40| = x − 40.

|x − 10| > 4|x − 40|

Substitute:

x − 10 > 4(x − 40)

Distribute:

x − 10 > 4x − 160

x > 4x − 150

Subtract 4x :

−3x > −150

Divide by −3 and “swap:”

x < 50

So this gives us 40 < x < 50. When we combine this with the solutions from CASE 2, we get 34 < x < 50.

Calculator

12 . 2 If |a − 5| = 7, then either a − 5 = 7 or a − 5 = − 7, so either a = 12 or a = −2. Since a < 0, a must be −2, and |−2| = 2.

13 . 8 CASE 1: If 6 − 3n is positive, then

|6 − 3n | = 6 − 3n , so

16 > 6 − 3n > 19

Subtract 6:

10 > −3n > 13

Divide by −3 and “swap:”

−10/3 < n < −13/3

But this contradicts the fact that n is positive.

CASE 2: If 6 − 3n is negative, then

|6 − 3n | = −(6 − 3n ), so

16 > −(6 − 3n ) > 19

Distribute:

16 > −6 + 3n > 19

22 > 3n > 25

Divide by 3:

7.33 > n > 8.33

And the only integer in this range is n = 8.

14 . 7

20 − 2n > 5

Subtract 20:

−2n > −15

Divide by −2 and “swap:”

n < 7.5

Multiply by 3:

2n > 12

Divide by 2:

n > 6

Since n must be an integer between 6 and 7.5, n = 7.

15 . 4.75 The distance from 3 to −1.5 is |3−(−1.5)| = 4.5. Therefore the two numbers that are 4.5 away from 9.25 are 9.25 + 4.5 = 13.75 and 9.25 − 4.5 = 4.75.

16 . ½ or .5 If the equation is true for all values of x , let”s choose a convenient value for x ,

like x = 1.

|2x + 1| = 2|kx |

Substitute x = 1:

|2(1) + 1| = 2|k − 1|

Simplify:

3 = 2|k − 1|

Divide by 2:

1.5 = |k − 1|

Therefore

±1.5 = k − 1

k = 2.5 or −0.5

Now try x = 0:

|2(0) + 1| = 2|k − 0|

Simplify:

1 = 2|k |

Divide by 2:

0.5 = |k |

Therefore

±0.5 = k

Therefore, k = −0.5 and so |k | = |−0.5| = 0.5.

17 . C Recall that the expression |x − 2| means “the distance from x to 2,” so the statement |x − 2| < 1 means “The distance from x to 2 is less than 1.” Therefore, the solution set is all of the numbers that are less than 1 unit away from 2, which are all the numbers between 1 and 3.

18 . C

Multiply by 2:

a + b > c + 2b

Subtract b :

a > c + b

19 . B The formal translation of this statement is |x − 1| > |x − 3|, which we can solve algebraically by considering three cases: (I) x ≤ 1, (II) 1 < x ≤ 3, and (III) x > 3, but it is probably easier to just graph the number line and notice that the midpoint between 1 and 3, that is, 2, is the point at which the distance to 1 and the distance to 3 are equal. Therefore, the points that are farther from 1 than from 3 are simply the points to the right of this midpoint, or x > 2.

20 . B

4x 2 ≥ 9

Take square root:

|2x | ≥ 3

If x > 0:

2x ≥ 3

Divide by 2:

x ≥ 1.5

If x < 0:

2x ≤ −3

Divide by 2:

x ≤ −1.5

21 . D Notice that the midpoint of the segment shown is 3, and the graph shows all points that are less than 3 units in either direction. Therefore, |x − 3| < 3.

22 . B (A) is untrue if x = 0, (C) is untrue for x = −2, and (D) is untrue if x = 0.5. But (B) is true for any real number.

Skill 4: Working with Linear Systems

Lesson 11: Constructing, graphing, and interpreting linear systems

A system of equations is just a set of equations that apply simultaneously to a given problem situation. Solving for the system means finding all sets of values for the unknowns that make all of the equations true. Systems of equations can be analyzed both algebraically (by exploring the equations) or geometrically (by exploring the graphs).

Two high school teachers took their classes on a field trip to a museum. One class spent \$154 for admission for 20 students and 3 adults, and the other class spent \$188 for admission for 24 students and 4 adults. Which of the following systems of equations could be solved to determine the price of a single student admission, s , and the price of a single adult admission, a , in dollars?

1. A) a+s = 51

44s + 7a = 342

1. B) 20s+ 3a= 154

24s + 4a = 188

1. C)
2. D) 20 + 24 =s

3 + 4 = a

(Medium ) This problem can be described with a two-by-two system of equations , that is, two equations with two unknowns. The two equations come from two facts: one class spent \$154 for admission and the other class spent \$188 for admission. The cost of 20 student admissions and 3 adult admissions is 20s + 3a , so the first equation is 20s + 3a = 154. Similarly, the equation for the other class is 24s + 4a = 188, so the correct answer is (B).

y = 2x − 3

y = −2x + 17

If the solutions to the two equations above are graphed in the xy -plane, what is the y -coordinate of the point at which the graphs intersect?

(Easy ) Since the equations of both lines are given in slope-intercept form, we could graph the two lines in the xy -plane to find their point of intersection.

Therefore, the point (5, 7) gives us the only solution to this system, and so the answer to the original question is 7.

Alternately, (as we will see in Lesson 13) we can just add the corresponding sides of the two equations together to get 2y = 14, which yields y = 7 .

The solution of a two-by-two system of equations can be visualized as the intersection of their graphs in the xy -plane.

If the graphs are parallel lines, or other non-intersecting graphs, then the system has no solution . If the graphs intersect multiple times, then the system has multiple solutions .

y − 4x = 6

16x = 4y + k

For what value of k does the system of equations above have at least one solution?

1. A) −32
2. B) −30
3. C) −24
4. D) −20

(Medium) This is a two-by-two system of linear equations, and so its solution is the intersection of those two lines. If we convert them to slope-intercept form, we get y = 4x + 6 and y = 4xk/ 4, which reveals that these two lines have the same slope. This means that they are either parallel lines or identical lines. Two lines with the same slope can intersect only if they are the same line, and therefore −k/ 4 = 6 and k = −24.

Lesson 12: Solving systems by substitution

Let”s go back to the second linear system from Lesson 11. This system can also be solved with a simple application of the Law of Substitution.

y = 2x − 3

y = −2x + 17

1. Substitute fory :

2x − 3 = −2x + 17

4x − 3 = 17

4x = 20

1. Divide by 4:

x = 5

1. Plug into either original equation to findy :

y = 2(5) − 3 or −2(5) + 17 = 7

When one of the equations in a system is already solved for one variable (or when it”s relatively easy to solve it for one variable), then substituting for this variable in the other equation often makes it easier to solve the system.

3x + y = 3y + 4

x + 4y = 6

Based on the system of equations above what is the value of the product xy ?

(Medium ) This system is not quite as tidy as the previous one, but we can still solve it by using the Law of Substitution.

3x + y = 3y + 4

x + 4y = 6

Subtract 4y from second equation to isolate x :

x = −4y + 6

Substitute for x in first equation:

3(−4y + 6) + y = 3y + 4

Simplify left side:

−11y + 18 = 3y + 4

14 = 14y

Divide by 14:

1 = y

Substitute y = 1 to find x :

x = −4(1) + 6 = 2

Evaluate xy :

xy = (2)(1) = 2

Lesson 13: Solving systems by linear combination

3x + 6y = 18

3x + 4y = 6

Based on the system of equations above, what is the value of y ?

(Easy ) Although this system can be solved by substitution (try it as an exercise), the setup of these equations suggests a much easier method, known as linear combination . It”s based on a simple idea:

The Law of Combination

If a = b and c = d , then a + c = b + d , a c = b d , and ac = bd

In other words, you should always feel free to add, subtract, or multiply the corresponding sides of two equations to make a new equation.

If we apply this rule to our system, notice that we can easily eliminate x from the system by just subtracting the equations:

Divide by 2:

y = 6

3xy = 20

2x + 4y = 7

Based on the system of equations above, what is the value of x − 5y ?

(Medium ) This question looks tougher than the previous one, because it”s not just asking for x or y . It seems that the question requires us to solve the system for x and y and then to plug these values into the expression x − 5y and evaluate. We could do that, but there is a much simpler method. Notice that a simple combination gives us the expression the question is asking for.

Subtract equations:

Using Linear Combination

When you”re given a system of equations on the SAT, always notice carefully what the question is asking you to evaluate . Even if it appears to be the value of a complicated expression, often you can find it with a simple combination of the given equations.

Exercise Set 5 (No Calculator)

1

If 3x + 2y = 72, and y = 3x , what is the value of x ?

2

If 2a − 7b = 10 and 2a + 7b = 2, what is the value of 4a 2 − 49b 2 ?

3

If the lines y = −4x − 3 and y = −3xb intersect at the point (−1, c ), what is the value of b ?

4

If the lines 4x + 5y = 13 and 4y + kx = 2 are parallel, what is the value of k ?

5

If the lines 4x + 5y = 13 and 6ykx = 6 are perpendicular, what is the value of k ?

6

Based on the system of equations above, what is the value of

7

If ab = −4 and abc = 12, what is the value of

8

If a and b are constants and the graphs of the lines 2x − 3y = 8 and ax + by = 2 are perpendicular, then what is the value of

9

5xy = 11

2x − 2y = 9

Based on the system of equations above, what is the value of 3x + y ?

1. A) −2
2. B) 0
3. C) 2
4. D) 4

10

Two numbers have a difference of 4 and a sum of −7. What is their product?

1. A) −33
2. B) −10.25
3. C) 8.25
4. D) 10.25

11

It costs Emma p dollars to make each of her custom bracelets, which she sells for m dollars apiece. She makes a profit of \$60 if she makes and sells 5 of these bracelets, but she only makes a profit of \$10 if she makes 5 bracelets but only sells 4 of them. How much does it cost Emma to make each bracelet?

1. A) \$36
2. B) \$38
3. C) \$48
4. D) \$50

Exercise Set 5 (Calculator)

12

If 2y = x + 1 and 4x + 6y = 0, then y =

13

If and , then y =

2x − 5y = 20

10x − 25y = 4k

14

For what value of k does the system of equations above have at least one solution?

15

At the beginning of the week, the ratio of cats to dogs at Glenna”s Pet Store was 4 to 5. By the end of the week, the number of cats had doubled, while the number of dogs had increased by 12. If the ratio of cats to dogs at the end of the week was 1 to 1, how many cats did the store have at the beginning of the week?

16

Jenny originally had twice as many friendship bracelets as Emilie. After Jenny gave Emilie 5 of her friendship bracelets, Jenny still had 10 more than Emilie. How many friendship bracelets did Jenny have originally?

17

The average (arithmetic mean) of x and y is 14. If the value of x is doubled and the value of y is tripled, the average (arithmetic mean) of the two numbers remains the same. What is the value of x ?

18

7m + 10n = 7

6m + 9n = 1

Based on the system of equations above, what is the value of 4m + 4n ?

19

In the xy -plane, perpendicular lines a and b intersect at the point (2, 2). If line a contains the point (7, 1), which of the following points is on line b ?

1. A) (0, 1)
2. B) (4, 5)
3. C) (7, 3)
4. D) (3, 7)

20

Which of the following pairs of equations has no solution in common?

1. A) 2x− 3y= 1 and 6x − 9y = 3
2. B) y= 4xand y = −4x
3. C) 2x− 3y= 1 and 6x − 9y = 2
4. D) y= 4xand 2y − 8x = 0

21

In the xy -plane, the line l is perpendicular to the line described by the equation . What is the slope of line l ?

1. A) −2
2. B)
3. C)
4. D) 2

No Calculator

1 . 8

3x + 2y = 72

Substitute y = 3x :

3x + 2(3x ) = 72

Simplify:

9x = 72

Divide by 9:

x = 8

2 . 20

4a 2 − 49b 2

Factor:

(2a − 7b )(2a + 7b )

Substitute:

(10)(2) = 20

3 . 2

y = −4x − 3

Substitute x = −1, y = c :

c = −4(−1) − 3

Simplify:

c = 1

Other equation:

y = −3xb

Substitute x = −1, y = 1:

1 = −3(−1) − b

Simplify:

1 = 3 − b

Subtract 3:

−2 = −b

Divide by −1:

2 = b

4 . 3.2 or 16/5 Parallel lines must have equal slopes. The slope of 4x + 5y = 13 is −4/5, and the slope of 4y + kx = 2 is −k /4.

Cross-multiply:

−5k = −16

Divide by −5:

k = 16/5 = 3.2

5 . 7.5 or 15/2 Perpendicular line have slopes that are opposite reciprocals. The slope of 4x + 5y = 13 is −4/5, and the slope of 6ykx = 6 is k /6.

Cross-multiply:

−4k = −30

Divide by −4:

k = 7.5

6 . .25 or ¼ First equation:

Divide by 2:

Second equation:

Subtract 1:

Reciprocate:

Multiply:

7 . .75 or ¾

abc = 12

Substitute ab = −4:

(−4)c = 12

Divide by −4:

c = −3

Expression to evaluate:

Substitute c = −3 and ab = −4:

8 . 4.5 or 9/2 The slope of 2x − 3y = 8 is 2/3, and the slope of ax + by = 2 is −a/b . If the two lines are perpendicular, then the slopes are

opposite reciprocals:

Reciprocate:

Multiply by 3:

9 . C

5xy = 11

2x − 2y = 9

Subtract equations:

3x + y = 2

10 . C

ab = 4

a + b = −7

2a = −3

Divide by 2:

a = −1.5

Substitute a = −1.5:

−1.5 + b = −7

b = −7 + 1.5 = −5.5

Evaluate product:

ab = (−1.5)(−5.5) = 8.25

11 . B Let c = the cost to make each one of Emma”s bracelets.

5m − 5c = 60

4m − 5c = 10

Subtract:

m = 50

Substitute m = 50

5(50) − 5c = 60

Simplify:

250 − 5c = 60

Subtract 250:

−5c = −190

Divide by −5:

c = 38

Calculator

12 . 2/7 or .286 or .285

2y = x + 1

Subtract 1:

2y − 1 = x

Given:

4x + 6y = 0

Substitute x = 2y − 1:

4(2y − 1) + 6y = 0

Distribute:

8y − 4 + 6y = 0

Simplify:

14y − 4 = 0

14y = 4

Divide by 14:

y = 4/14= 2/7

13 . 1/6 or .166 or .167

12x = 2

Divide by 12:

x = 2/12 = 1/6

14 . 25 The slope of 2x − 5y = 20 is 2/5. The slope of 10x − 25y = 4k is 10/25 = 2/5. Since the two lines have the same slope, they have no points of intersection unless they are the same line.

2x − 5y = 20

10x − 25y = 4k

Multiply first equation by 5:

10x − 25y = 100

Therefore, 4k = 100 and so k = 25.

15 . 16 If the original ratio of cats to dogs is 4 to 5, then we can say there were 4n cats and 5n dogs to start. At the end of the week, therefore, there were 8n cats and 5n + 12 dogs. If this ratio was 1:1, then

8n = 5n + 12

Subtract 5n :

3n = 12

Divide by 3:

n = 4

Therefore, there were 4n = 4(4) = 16 cats at the beginning of the week.

16 . 40 Let x = the number of friendship bracelets Emilie had to start. This means that Jenny originally had 2x bracelets. After Jenny gave 5 of them to Emilie, Jenny had 2x − 5 and Emilie had x + 5. If Jenny still had 10 more than Emilie, then

2x − 5 = 10 + (x + 5)

Simplify:

2x − 5 = x + 15

x = 20

This means that Jenny had 2x = 2(20) = 40 to start.

17 . 56

Multiply by 2:

x + y = 28

If x is doubled and y is tripled, the average remains the same:

Multiply by 2:

2x + 3y = 28

Previous equation:

x + y = 28

Multiply by 3:

3x + 3y = 84

Other equation:

2x + 3y = 28

Subtract equations:

x = 56

18 . 24

7m + 10n = 7

6m + 9n = 1

Subtract equations:

m + n = 6

Multiply by 4:

4m + 4n = 24

19 . D Line a contains the points (2, 2) and (7, 1); therefore, it has a slope of . If line b is perpendicular to line a , then it must have a slope of 5 (the opposite reciprocal of −1/5). You might find it helpful to sketch the line with slope 5 through the point (2, 2), and confirm that is passes through the point (3, 7), which is one unit to the right and one 5 units up.

20 . C In order for two lines in the xy -plane to have no points in common, they must be parallel and nonidentical. The only two such lines among these choices are 2x − 3y = 1 and 6x − 9y = 2, which both have a slope of 2/3, but have different y -intercepts of −1/3 and − 2/9.

21 . A

Multiply by 2xy :

2y + x = 2x

Subtract x :

2y = x

Divide by 2:

This line has a slope of 1/2, so the perpendicular must have a slope of −2.

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