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8. How many ounces of chocolate that is 60% cocoa should be mixed with 4 oz of chocolate that is 80% cocoa to produce a mixture that is 75% cocoa?

9. At precisely noon, one plane leaves New York, heading for Orlando, and another leaves Orlando, heading for New York. The distance from New York to Orlando is 1300 mi. The plane from New York flies at 450 mph and the Orlando plane flies at 490 mph. When will the planes be 125 mi apart?

10. I leave my family’s vacation cabin at 8 a.m. and start driving home at a nice, safe 45 mph. Two hours later, my husband, who always drives as fast as the law will allow, leaves the cabin and starts driving home at 65 mph. When can I expect him to pass me?

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•3•

Linear inequalities

When you solve an equation, you find the value of the variable that makes the two sides of the equation identical. Inequalities ask you to find the values that make one side larger than the other. The solution will be a set of numbers, rather than the single value that solves a linear equation or the pair of values that solve an absolute value equation.

Simple inequalities

Linear inequalities can be solved in much the same way as linear equations with one important exception. When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign reverses. Remember that the positive and negative sides of the number line are mirror images of one another. When you multiply both sides by a negative, you go through the looking glass and the larger-smaller relationship flips.

The solution set of an inequality can be graphed on the number line by shading the appropriate portion of the line as shown in Figure 3.1. Use an open circle to mark the boundary of the solution set if the inequality sign is < or >, and a solid dot for inequalities containing, ≤ or ≥.

Figure 3.1 Solutions of inequalities graphed on the number line.

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EXERCISE 3.1

Solve each inequality and graph the solution set on a number line.

1. 3x −5 ≥ 22

2. 2x − 5 > 13 − 4x

3. 3x + 2 ≤ 8 x + 22

4. 12x + 3 < x + 36

5. t − 9 ≥ 24 − 10t

6. 2y − 13 > 4(2 − y)

7. 5x + 10(x − 1) ≥ 95

8. 5x − 4 ≤ 13x + 28

9. 3x − 2 < 2x − 3

10. −x + 5 ≥ −2 + x

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Compound inequalities

A compound inequality can be written as two linear inequalities joined by the conjunction and or the conjunction or. Some and inequalities can be written in a condensed form; for example, the compound inequality 7 > 5x + 2 and 5x + 2 > −13 can be written as 7 > 5x + 2 > −13. To solve compound inequalities, rewrite them as two inequalities, solve each inequality separately, and join the two solutions with the same conjunction.

Figure 3.2 Graphing the solution set of a compound inequality.

To graph the solution set of a compound inequality, graph the solution sets of the two component inequalities, shading lightly. If the inequalities are connected with the conjunction and, the solution set is the intersection or overlap of the two graphs. For compound inequalities that are connected by or, you want to keep both areas of shading. If the two shaded areas for an or inequality overlap, you may find that you have only one shaded area(see Figure 3.2).

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EXERCISE 3.2

Solve each inequality and graph the solution set on a number line.

1. x − 10 > 30 or x − 4 < − 10

2. 6 < y − 9 < 15

3. −3 ≤ 4x + 5 < 2

4. 6x − 4 ≥ 26 or 3x + 8 < 14

5. −43 < 11x + 1 ≤ 12

6. 8 + 3x > 4 − 3x or 9x + 12 < −87 − 2x

7. −10 ≤ 15y + 5 ≤ 6

8. −13 < 47 − 3x < −2

9. 3x + 4 < 6x + 7 or 5x − 2 > 3x + 18

10. −4y + 31 < y + 16 ≤ 3y + 2

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Absolute value inequalities

In an absolute value equation, you look for two solutions. You consider the possibility that the expression within the absolute value signs is equal to the number on the other side of the equation and the possibility that the expression is equal to the opposite of that number. In an absolute value inequality, you also have