Idiot's Guides: Algebra I (2015)
Part II. Linear Relationships
Chapter 5. Inequalities
In This Chapter
· The similarities and differences between equations and inequalities
· How to solve single inequalities
· Decomposing compound inequalities
· Representing solutions of inequalities on the number line
Like any language, algebra needs to be able to say different things. Expressions are the phrases of mathematical communication, and equations are one type of algebraic sentence. Equations say two expressions have the same value, but that’s not always the case. Sometimes the values of the two expressions are unequal with one larger than the other. In this chapter, we’ll look at statements of inequality, and see what information we can gain by solving them.
Types of Inequalities
Inequalities are algebraic sentences that tell you that one expression is larger or smaller than another. There are several different symbols that are used to communicate inequality. The simplest, but least commonly used, is the “not equal” sign: ≠. If you see x ≠ 5, which is read “x is not equal to 5,” you know that x is not the same as 5. That gives you some information, but not a lot, which probably explains why that symbol isn’t used more.
DEFINITION
An inequality compares two expressions that are not equal and shows which one is larger.
The more common inequality symbols are < for “is less than” and > for “is greater than.” At times, you want to indicate that the upper or lower limit for the value of a variable is itself an acceptable value. In that case, you can add a small line segment that looks like part of an equal sign under the inequality sign. The symbol ≤ says “is less than or equal to” and the symbol ≥ says “is greater than or equal to.”
This chart shows examples of the language that translates to inequalities.
x is less than 12. |
x < 12 |
y is greater than 9. |
y > 9 |
z exceeds 7. |
z > 7 |
t is no more than 20. |
t ≤ 20 |
n is at most 13. |
n ≤ 13 |
w is no less than 15. |
w ≥ 15 |
a is at least 11. |
a ≥ 11 |
CHECK POINT
Write an inequality for each sentence.
1. A number, x, is less than 12.
2. The interest rate, r, does not exceed 4%.
3. In order to ride the roller coaster, your height, h, must exceed 48 inches.
4. The price of a ticket, p, is at least $20.
5. Refunds are available for those whose income, I, is no more than $12,000.
Write a sentence for each inequality.
6. x ≥ 16
7. y < 12
8. t > 22
9. z ≤ 100
10. a > 0
Solving Inequalities
The solution of an inequality is not a single number. It’s a range, or set, of numbers. Luckily, the process of finding that solution set is very similar to the process of solving an equation. In fact, there’s only one significant difference.
To solve the inequality 3x − 5 < 2x + 12, you can take the same steps you’d take to solve the equation 3x − 5 = 2x + 12. You want to eliminate one of the variable terms by subtracting from both sides. 3x − 5 < 2x + 12 becomes 3x − 5 − 2x < 2x + 12 − 2x or x − 5 < 12. Adding 5 to both sides tells you that x < 17. You now have a range of values for the variable.
ALGEBRA TRAP
Unlike equations, inequalities are not symmetric. The equation 3x − 5 = 2x + 12 can be rewritten as 2x + 12 = 3x − 5, but 3x − 5 < 2x + 12 and 2x + 12 < 3x − 5 do not say the same thing at all. In fact, they're opposites. If you must rewrite an inequality, make sure that the smaller expression is at the smaller end of the inequality sign. 3x − 5 < 2x + 12 is equivalent to 2x + 12 > 3x − 5, not to 2x + 12 < 3x − 5.
The solution to this inequality is not a single number. It’s not 17, but it could be any number less than 17. You can test any number less than 17 to check. If I pick 2, 3 · 2 − 5 is 1, and 2 · 2 + 12 is 16, and 1 is less than 16. Replacing x with 2, or any number less than 17, will make the inequality statement true.
TIP
Look carefully at the inequality sign. If the sign is < or > the number on the other side is not part of the solution set. If the sign has an “or equal to,” as in ≤ or ≥, then the number is part of the solution set.
Let’s look at an inequality for which you need to do something different. Take the inequality −2x + 5 > 3. Start solving it just as if it were an equation, by subtracting 5 from both sides.
−2x + 5 > 3
−2x + 5 − 5 > 3 − 5
−2x > − 2
Now take a minute to play guess and test, and find a value forx that makes the statement true. You’re dealing with negative numbers, so remember that you have to think about left and right on the number line. If you try x = 2, the inequality becomes − 2 · 2 > − 2, which simplifies to −4 > −2, but that’s not true. If you try x = −2, the inequality becomes −2(−2) > −2, which is 4 > −2, a true statement. So x = 2 is not part of the solution but x = −2 is. Remember that and let’s take another step in solving.
If you were solving an equation, you would divide both sides by -2. Let’s try that.
That says that the solution is all numbers greater than 1, but you already know that 2, which is greater than 1, doesn’t work and -2, which is less than 1, does. What went wrong?
When you divided by a negative number, you bumped into that “through the looking glass” appearance of negative numbers on the number line. When you solve an equation, you’re looking for one number, so the mirroring doesn’t matter. When you solve an inequality, the number you find is just a divider and you need to know which side of that border you want to be on. To get that straight you need to add one extra rule for inequalities: when you multiply or divide by a negative number, reverse the direction of the inequality sign.
The correct solution of −2x + 5 > 3 will look like this.
To solve an inequality, follow these steps.
1. Simplify the left side and the right side so that there are no more than two terms on either side.
2. If there are variable terms on both sides, eliminate one of them by adding or subtracting the same term to both sides.
3. Isolate the variable by performing inverse operations.
4. If you multiply or divide both sides of the inequality by a negative number, reverse the direction of the inequality sign.
5. Graph the solution set on a number line.
6. Test a value from your solution set in the original inequality.
To solve the inequality 3(x − 3) + 1 ≤ 4(x − 6) − 2, first simplify the left side.
3(x− 3) + 1 ≤ 4(x− 6)− 2
3x− 9 + 1 ≤ 4(x− 6)− 2
3x− 8 ≤ 4(x− 6)− 2
Then simplify the right side.
3x− 8 ≤ 4(x− 6)− 2
3x − 8 ≤ 4x − 24 − 2
3x − 8 ≤ 4x − 26
Eliminate one of the variable terms by subtracting 4x from both sides.
3x − 8 ≤ 4x − 26
3x − 8 − 4x ≤ 4x − 26 − 4x
−x − 8 ≤ − 26
Add 8 to both sides.
− x − 8 ≤ − 26
− x − 8 + 8 ≤ − 26 + 8
− x ≤ −18
Divide both sides by -1 and reverse the inequality sign.
Graph the solution on a number line.
Test a value from the solution set in the original inequality. Pick a number larger than 18, like 20, and substitute that for x in the original inequality.
3(x − 3) + 1 ≤ 4(x− 6) − 2
3(20 − 3) + 1 ≤ 4(20 − 6) − 2
3(17) + 1 ≤ 4(14)− 2
51 + 1 ≤ 56 − 2
52 ≤ 54
The result after simplifying is a true statement—52 is truly less than 54—so you can be confident that your solution is correct.
CHECK POINT
Solve each inequality.
11. x + 7 > 4
12. y − 7 < −1
13. 2x − 3 ≥ 5
14. 5 − 3t < 23 + t
15. 3y + 11 ≤ 17 − y
For each inequality, tell whether the indicated value of the variable is part of the solution.
16. −7x − 9 ≥ 12, x = −4
17. 5 − x > 11, x = −6
18. 2y + 3 < 4, y = 1
19. 6 − 3x ≤ − 12, x = 5
20. 11t + 19 ≤ 8t + 61, t = 10
Picturing Solutions on a Number Line
Because the solution of an inequality is not a single number but a whole set of numbers, it’s helpful to have a visual representation of the solution. You can create that picture by graphing the solution set on a number line.
To graph the solution of an inequality on the number line, you first need to locate that number that divides the numbers that are part of the solution from those that are not. If your solution is x ≥ − 3, -3 is the dividing point.
TIP
Plan ahead when graphing the solution of an inequality on a number line. The number line goes on forever, so you can’t show all of it, and you can choose what part and how much you want to show. Your tick marks and labels don’t need to count by ones either. If you’re graphing x ≥ 150, you don’t want to start at 0 and count by ones. Choose the range and scale that make sense for your task.
Before you mark that number, you must decide whether that border number is or is not part of the solution. If the inequality is ≥ or ≤, the number that divides the line is part of the solution. Use a solid dot to mark that point.
If the symbol is < or >, the number is not part of the solution. You still need to mark the number, but do it by just putting an open circle. If you wanted to show x>-3 instead of x ≥ − 3, you’d mark -3 like this.
Once you’ve marked the dividing point, you have to show whether the solution falls above or below that. You do this by shading the side that is the solution, and placing an arrow on the line of shading to show that it goes on forever. For x ≥ −3, it looks like this.
CHECK POINT
Graph each inequality on a number line.
21. x > 5
22. t ≥ −8
23. y < −4
24. z ≤ 7
25. a ≤ 9
Solve each inequality and graph the solution on a number line.
26. 3x − 7 ≤ 5
27. x − 6 > 5x + 2
28. −2x + 1 ≥ 9
29. − 6y + 5 < 3y + 14
30. 5(t − 2) ≤ (2t − 3) + (t − 7)
Compound Inequalities
Solving an inequality is just like solving an equation except that you reverse the inequality sign if you multiply or divide both sides by a negative number. The solution to an inequality is a set of numbers, all the numbers above or below a certain boundary point. Sometimes the boundary is part of the solution and sometimes it is not.
Having such a large set of solutions can be unsatisfying. Although you can’t get a specific number, you can narrow things down. Compound inequalities are one way we can do that. A compound inequality is a statement made up of two inequalities connected either with the word or or the word and.
DEFINITION
A compound inequality is actually two inequalities connected by either the word and or the word or. If the and is used, the compound inequality is called a conjunction. If the pieces are joined by or, it’s called a disjunction.
A disjunction, or or inequality, might look like 7x + 5 ≤ − 16 or 11 − 3x < − 1. It truly is just two inequalities, using the same variable, connected by the word “or.” To solve the compound statement, solve each of the individual inequalities and connect the solutions with an “or.” (Don’t forget to reverse the inequality sign if you divide by a negative.)
Now you know that the numbers that meet the description in this compound statement are -3, all the numbers less than -3 and all the numbers greater than 4. Turning that around, you could say that the numbers that are not in the solution are numbers greater than -3 and less than or equal to 4. Here’s what it looks like on the number line.
The solution to an or inequality usually looks like two arrows pointing away from one another, with a space between them. If you find that the two arrows overlap, the solution is all real numbers.
The or inequality gives a little more information about the solution, but usually leaves us with a very large set. The and inequality generally restricts the solution set more effectively, but it takes an extra step or two to solve. Consider the compound inequality 7 ≤ 2x − 3 < 11. The and inequality is often presented in this condensed form. You don’t see the word and, as the two inequalities are compressed into what looks like one and a half.
The inequality 7 ≤ 2x − 3 < 11 is actually 7 ≤ 2x − 3 and 2x − 3 < 11, two inequalities, with the same variable, connected by the word “and.” What allows them to be compressed is that they share the expression 2x − 3 and both inequality signs point in the same direction. The statement 7 ≤ 2x − 3 < 11 says that 2x − 3 is between 7 and 11. It’s at least 7, but less than 11.
To solve a compound inequality like this one, first break it into its two inequalities. The first inequality starts at the beginning of the statement, and stops just before the second inequality sign. . The second inequality begins just after the first inequality sign and goes to the end of the statement. . Connect the inequalities with the word “and” and then solve each separately. After you’ve solved each inequality, you usually can compress the two solutions into a single statement.
The graph of this solution set has a solid dot at 5, an open circle at 7, and shades all the numbers in between.
Don’t try to force the compression of compound inequalities. They must be and inequalities, the inequality signs must point in the same direction, and there must be a shared expression that fits in the middle. You can compress 5 < 3x + 1 and 3x + 1 ≤ 12 into 5 < 3x + 1 ≤ 12, but you can’t compress 5 > 3x + 1 with 3x + 1 ≤ 12, or 3x + 1 < 5 with 3x + 1 ≤ 12.
THINK ABOUT IT
The solution of an and inequality can usually be condensed, if the original was condensed. Your two separate solutions will have the form a < x and x < b, with a < b. If a > b, however, you have a contradiction. It’s possible to have 3 < x < 5 but it’s not possible to have 5 < x < 3. If x is greater than 5, it can’t be less than 3. If you find yourself in this situation, first check to see if you divided by a negative and forgot to reverse the inequality sign. If there’s no mistake, the inequality has no solution.
The compound inequality −5 ≤ 7 − 3x ≤ 43 can be separated into two inequalities connected by an and.
Dividing both sides by -3 caused the inequality sign to change from ≤ to ≥. The solution, presented from large to small, 4 ≥ x ≥ − 12, is correct and acceptable, but you might want to carefully rewrite it from small to large, − 12 ≤ x ≤ 4, just because that’s the more familiar form. Either way, it looks like this when graphed.
CHECK POINT
Solve each compound inequality. Graph the solution set on the number line.
31. 4y − 7 > 5 or 5y + 7 < −3
32. −1 ≤ 6z − 7 ≤ 11
33. 3x − 8 ≤ 4x − 26 or x + 5 < 12
34. 2 < 3t − 7 < 5
35. 17 < 4 y + 5 ≤ 1
36. 7t − 3 ≤ 8t + 5 or 4t + 1 > 2t − 9
37. −11 ≤ 3x + 1 ≤ 19
38. 9 < 2x − 1 or 2x − 1 < 6
39. 21 + 1 < 5 − 3t ≤ 20
40. y − 2 ≤ 5(y − 2) < (y − 2) + 8
The Least You Need to Know
· Solve inequalities as if they are equations, but reverse the inequality sign if you multiply or divide by a negative number.
· Compound inequalities are two inequalities connected by “or” or “and.” Solve each of the inequalities separately.
· Graph the solution set of an inequality on a number line.