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# 2.8E: Exercises


### Practice Makes Perfect

Solve Absolute Value Equations

In the following exercises, solve.

ⓐ $$|x|=6$$ ⓑ $$|y|=−3$$
ⓒ $$|z|=0$$

ⓐ $$|x|=4$$ ⓑ $$|y|=−5$$
ⓒ $$|z|=0$$

Answer

ⓐ $$x=4,x=−4$$ ⓑ no solution ⓒ $$z=0$$

ⓐ $$|x|=7 ⓑ \(|y|=−11$$
ⓒ $$|z|=0$$

ⓐ $$|x|=3$$ ⓑ $$|y|=−1$$
ⓒ $$|z|=0$$

Answer

ⓐ $$x=3,x=−3$$ ⓑ no solution ⓒ $$z=0$$

$$|2x−3|−4=1$$

$$|4x−1|−3=0$$

Answer

$$x=1,x=−\frac{1}{2}$$

$$|3x−4|+5=7$$

$$|4x+7|+2=5$$

Answer

$$x=−1,x=−\frac{5}{2}$$

$$4|x−1|+2=10$$

$$3|x−4|+2=11$$

Answer

$$x=7,x=1$$

$$3|4x−5|−4=11$$

$$3|x+2|−5=4$$

Answer

$$x=1,x=−5$$

$$−2|x−3|+8=−4$$

$$−3|x−4|+4=−5$$

Answer

$$x=7,x=1$$

$$|34x−3|+7=2$$

$$|35x−2|+5=2$$

Answer

no solution

$$|12x+5|+4=1$$

$$|14x+3|+3=1$$

Answer

no solution

$$|3x−2|=|2x−3|$$

$$|4x+3|=|2x+1|$$

Answer

$$x=−1,x=−\frac{2}{3}$$

$$|6x−5|=|2x+3|$$

$$|6−x|=|3−2x|$$

Answer

$$x=−3,x=3$$

Solve Absolute Value Inequalities with “less than”

In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.

$$|x|<5$$

$$|x|<1$$

Answer

$$|x|\leq 8$$

$$|x|\leq 3$$

Answer

$$|3x−3|\leq 6$$

$$|2x−5|\leq 3$$

Answer

$$|2x+3|+5<4$$

$$|3x−7|+3<1$$

Answer

$$|4x−3|<1$$

$$|6x−5|<7$$

Answer

$$|x−4|\leq −1$$

$$|5x+1|\leq −2$$

Answer

Solve Absolute Value Inequalities with “greater than”

In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.

$$|x|>3$$

$$|x|>6$$

Answer

$$|x|\geq 2$$

$$|x|\geq 5$$

Answer

$$|3x−8|>−1$$

$$|x−5|>−2$$

Answer

$$|3x−2|>4$$

$$|2x−1|>5$$

Answer

$$|x+3|\geq 5$$

$$|x−7|\geq 1$$

Answer

$$3|x|+4\geq 1$$

$$5|x|+6\geq 1$$

Answer

In the following exercises, solve. For each inequality, also graph the solution and write the solution in interval notation.

$$2|x+6|+4=8$$

$$|3x−4|\geq 2$$

Answer

$$x=4,x=27$$

$$|6x−5|=|2x+3|$$

$$|4x−3|<5$$

Answer

$$x=3,x=2$$

$$|2x−5|+2=3$$

$$|3x+1|−3=7$$

Answer

$$x=3,x=−\frac{11}{3}$$

$$|7x+2|+8<4$$

$$5|2x−1|−3=7$$

Answer

$$x=\frac{3}{2},x=−\frac{1}{2}$$

$$|x−7|>−3$$

$$|8−x|=|4−3x|$$

Answer

Solve Applications with Absolute Value

In the following exercises, solve.

A chicken farm ideally produces 200,000 eggs per day. But this total can vary by as much as 25,000 eggs. What is the maximum and minimum expected production at the farm?

An organic juice bottler ideally produces 215,000 bottle per day. But this total can vary by as much as 7,500 bottles. What is the maximum and minimum expected production at the bottling company?

Answer

The minimum to maximum expected production is 207,500 to 2,225,000 bottles

In order to insure compliance with the law, Miguel routinely overshoots the weight of his tortillas by 0.5 gram. He just received a report that told him that he could be losing as much as \$100,000 per year using this practice. He now plans to buy new equipment that guarantees the thickness of the tortilla within 0.005 inches. If the ideal thickness of the tortilla is 0.04 inches, what thickness of tortillas will be guaranteed?

At Lilly’s Bakery, the ideal weight of a loaf of bread is 24 ounces. By law, the actual weight can vary from the ideal by 1.5 ounces. What range of weight will be acceptable to the inspector without causing the bakery being fined?

Answer

The acceptable weight is 22.5 to 25.5 ounces.

### Writing Exercises

Write a graphical description of the absolute value of a number

In your own words, explain how to solve the absolute value inequality, $$|3x−2|\geq 4$$.

Answer

Answers will vary.

### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

This page titled 2.8E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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