Q: Which of the following is the correct solution set for $$|2x - 1| \geq 5$$?

$$x \leq -2$$ \,\text{or}\, x \geq 3$

Q: A chemical process requires the pH to be within 0.5 units of 7. What is the range of acceptable pH values?

$$6.5 \u003c \text{pH} \u003c 7.5$$

Q: Solve the equation $$|4x + 7| = 3.$$

$$x = -2.5$$ \,\text{or}\, x = -1$

Q: Solve the inequality $$|x| \geq 4$$ and express the solution set.

$$x \leq -4$$ \,\text{or}\, x \geq 4$

Question 1

Which of the following is the correct solution set for the equation $$|x - 4| = 7$$?

Accepted Answer

$$x = -3 or$$ $$x = 11$$

Question 2

If $$|2x + 5| = 0$$, what is the value of $$x$$?

Accepted Answer

$$x = -2.5$$

Question 3

Which of the following statements is true about the equation $$|X| = a$$ when $$a \u003c 0$$?

Accepted Answer

There is no solution; the solution set is the empty set.

Question 4

Solve the inequality $$|x + 2| \u003c 6.$$

Accepted Answer

$$-6 \u003c x + 2 \u003c 6$$, so $$-8 \u003c x \u003c 4$$

Question 5

A medical thermometer reads $$|T - 98.6| \u003e 1.4 to $$indicate a fever. What temperature values $$T$$ indicate a fever?

Accepted Answer

$$T  100$$

Question 6

Solve the equation $$|3x - 9| = 0$$ and interpret the solution in terms of distance on a number line.

Accepted Answer

$$x = 3$$, which is exactly 0 units from 9/3 = 3$

Question 7

Which of the following is the correct solution set for $$|2x - 1| \geq 5$$?

Accepted Answer

$$x \leq -2$$ \,	ext{or}\, x \geq 3$

Question 8

A chemical process requires the pH to be within 0.5 units of 7. What is the range of acceptable pH values?

Accepted Answer

$$6.5 \u003c 	ext{pH} \u003c 7.5$$

Question 9

Solve the inequality $$|x - 4| \u003e 9.$$

Accepted Answer

$$x  13$$

Question 10

Which of the following is the correct solution set for $$|x + 1| \leq 4$$?

Accepted Answer

$$-5 \leq x \leq 3$$

Question 11

A car rental company charges a fee based on the absolute value of the difference between the actual miles driven and the estimated miles: $$|m - 100|. If $$the fee is $$0$$ when the actual miles equals the estimate, what is the value of $$m$$?

Accepted Answer

$$m = 100$$

Question 12

Solve the equation $$|4x + 7| = 3.$$

Accepted Answer

$$x = -2.5$$ \,	ext{or}\, x = -1$

Question 13

Which of the following is the correct solution set for $$|x| \u003c 8$$?

Accepted Answer

$$-8 \u003c x \u003c 8$$

Question 14

Solve the inequality $$|2x + 3| \leq 7.$$

Accepted Answer

$$-7 \leq 2x + 3 \leq 7$$, so $$-5 \leq x \leq 2$$

Question 15

A factory requires the temperature to stay within 2 degrees of 70°F for optimal operation. Express the acceptable temperature range using absolute value notation.

Accepted Answer

$$|T - 70| \u003c 2$$

Question 16

Solve the equation $$|x + 5| = |x - 3|.$$

Accepted Answer

$$x = -1$$

Question 17

Which of the following is the correct solution set for $$|x - 2| \geq 0$$?

Accepted Answer

All real numbers $$x$$

Question 18

Solve the inequality $$|x| \geq 4$$ and express the solution set.

Accepted Answer

$$x \leq -4$$ \,	ext{or}\, x \geq 4$

Question 19

A student claims that the equation $$|x| = -2$$ has two solutions: $$x = 2$$ and $$x = -2. Is $$this correct?

Accepted Answer

No, there is no solution because absolute value cannot be negative.

Solving Equations and Inequalities with Absolute Value (Section 3.5)

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