SAT Test Prep

CHAPTER 8
ESSENTIAL ALGEBRA I SKILLS

Lesson 6: Inequalities, Absolute Values, and Plugging In

Inequalities as Unbalanced Scales


Inequalities are just unbalanced scales. Nearly all of the laws of equality pertain to inequalities, with one exception. When solving inequalities, keep the direction of the inequality (remember that “the alligator < always eats the bigger number”) unless you divide or multiply by a negative, in which case you “switch” the inequality.


Example:

Solve  for x.

You might be tempted to divide both sides by x and get , but this incorrectly assumes that x is positive. If x is positive, then , but if x is negative, then . (Switch the inequality when you divide by a negative!) But of course any negative number is less than 6, so the solution is either  or . (Plug in numbers to verify!)

Absolute Values as Distances


The absolute value of x, written as , means the distance from x to 0 on the number line. Since distances are never negative, neither are absolute values. For instance, since –4 is four units away from 0, we say .



The distance between numbers is found from their difference. For instance, the distance between 5 and –2 on the number line is . But differences can be negative, and distances can’t! That’s where absolute values come in. Mathematically, the distance between a and b is .


Example:

Graph the solution of .

You can think about this in two ways. First think about distances.  is the same as , which is the distance between x and –2. So if this distance must be greater than or equal to 3, you can just visualize those numbers that are at least 3 units away from –2:

Or you can do it more “algebraically” if you prefer. The only numbers that have an absolute value greater than or equal to 3 are numbers greater than or equal to 3 or less than or equal to –3, right? Therefore, saying  is the same as saying  or . Subtracting 2 from both sides of both inequalities gives  or , which confirms the answer by the other method.

Plugging In

After solving each of the examples above, you should, as with all equations and inequalities, plug in your solution to confirm that it works in the equation or inequality. But plugging in can also be a good way of solving multiple-choice problems that ask you to find an expression with variables rather than a numerical solution.


If a multiple-choice question has choices that contain unknowns, you can often simplify the problem by just plugging in values for the unknowns. But think first: in some situations, plugging in is not the simplest method.


Example:

If  , which of the following expresses r in terms of y and z?

If you pick r to be 6—it can be whatever you want, so pick an easy number!—then y is  and z is . The question is asking for an expression for r, so look for 6 among the choices. Plugging in your values gives (A) 10 (B) 11 (C) 12 (D) 5 (E) 6. Always evaluate all the choices because you must work by process of elimination. Only (E) gives 6, so it must be the right answer!

Concept Review 6: 
Inequalities, Absolute Values, and Plugging In

Express each of the following statements as equations or inequalities using absolute values.

Graph the solution to each of the following inequalities on the given number line. Check your answer by testing points.

Solve the following problem by plugging in, then see if you can solve it “algebraically.”

11. If  and , then which of the following expressions is equivalent to a?

SAT Practice 6:
Inequalities, Absolute Values, and Plugging In

1. If , then which of the following could NOT be the value of x?

(A) –5

(B) –4

(C) –3

(D) –2

(E) –1

2. If , and , then which of the following expressions must be positive?

(D) xy2

(E) xz2

3. Which of the following is equivalent to the statement ?

4. If , then which of the following represents all possible values of m?

(E) all real numbers

5. If  and , then what is the value of r-w in terms of a?

(A) 28 a

(C) 3 a

6. If x is the average (arithmetic mean) of k and 10 and y is the average (arithmetic mean) of k and 4, what is the average of x and y, in terms of k?

(D) 7 k

(E) 14 k

7. If  and , which of the following expresses x in terms of m and n?

8. What is the only integer n such that and 

9. If  and , then which of the following expresses a in terms of b and c?

(A) I only

(B) II only

(C) I and II only

(D) I and III only

(E) I, II, and III

10. Which of the following is equivalent to the statement “The distance from 1 to x is greater than the distance from 3 to x?”

(A) I only

(B) I and II only

(C) II and III only

(D) I and III only

(E) I, II, and III

Answer Key 6: 
Inequalities, Absolute Values, and Plugging In

Concept Review 6

But this is impossible, so there’s no solution!

11. (D) If you plug in , then  and . Since you’re looking for an expression that equals a, plug these into the choices and see which one gives :

Since (D) is the only choice that gives 4, it is the right choice. To solve it algebraically, solve each equation for a:

SAT Practice 6

1. A , which is not less than 20.

2. C To satisfy the inequalities, x must be negative, y must be negative, and z must be positive. You might choose , and  to confirm that (C) is the only one that gives a positive value.

4. E All absolute values are greater than or equal to zero, so any value of m would satisfy .

5. B You can solve by plugging in for the unknowns, but be careful to choose values that work in the equation. The simplest values that work are , and . In this case, . If you plug  into the choices, (B) is the only one that equals 28. Or you can solve algebraically by expressing r and w in terms of  and , so

6. C You might plug in . Since x is the average of k and 10, . Since y is the average of k and 4, . The average of x and y, then, is . If you then plug k = 2 into the choices, (C) is the only choice that equals 4.5.

7. B Plug in . Then  and . The question asks for an expression that equals x, so look for 3 in the choices when you plug in  and . The only choice that gives you 3 is (B)

The greatest integer n could be, then, is 7. Notice that 7 also satisfies the other inequality: , which of course is greater than 4.

9. C Plugging in isn’t good enough here, because more than one expression may be correct. The best method is substitution, using  and :

10. D The distance from 1 to x is  and the distance from 3 to x is , so I is clearly correct. To see why III is true, notice that 2 is the only number equidistant from 1 and 3, so all numbers that are farther from 1 than from 3 are greater than 2.