Question 1

Which of the following best describes the absolute value of a real number $$x$$?

Accepted Answer

The distance of $$x$$ from zero on the number line.

Question 2

Solve the equation $$8|x-3| = 4.$$

Accepted Answer

$$x = 3.5 or$$ $$x = 2.5$$

Question 3

Which of the following is the correct solution set for $$|4x| \u003c 12 in $$interval notation?

Accepted Answer

$$(-3, 3)$$

Question 4

If $$|3x-6| + 5 = 3$$, what is the solution for $$x$$?

Accepted Answer

No solution

Question 5

Which of the following is equivalent to $$|x| = a$$ for $$a \u003e 0$$?

Accepted Answer

$$x = a or$$ $$x = -a$$

Question 6

Solve the inequality $$|5x+1| \u003c 8$$ and express the answer in interval notation.

Accepted Answer

$$\left(-\frac{9}{5}, \frac{7}{5}\right)$$

Question 7

A machine is designed to produce rods of length 10 cm with a maximum allowable error of 0.03 cm. Which absolute value inequality represents the acceptable lengths $$L of $$the rods?

Accepted Answer

$$|L - 10| \leq 0.03$$

Question 8

Solve $$5 - |x+2| = 3.$$

Accepted Answer

$$x = 0 or$$ $$x = -4$$

Question 9

Suppose you want your room temperature to stay within 4 degrees of 70°F. Which absolute value inequality models this situation?

Accepted Answer

$$|T - 70| \leq 4$$

Question 10

Solve the inequality $$|5x+1| \u003e 8$$ and write the solution in interval notation.

Accepted Answer

$$(-\infty, -\frac{9}{5}) \cup (\frac{7}{5}, \infty)$$

Question 11

Which of the following is the correct first step when solving an equation involving absolute value?

Accepted Answer

Isolate the absolute value expression.

Question 12

If $$|x|  0$$, what is the equivalent compound inequality?

Accepted Answer

$$-a \u003c x \u003c a$$

Question 13

Which of the following statements is true about the equation $$|3x-6| = -2$$?

Accepted Answer

There is no solution because the absolute value cannot be negative.

Question 14

A student solves $$|x-3| = \frac{1}{2}$$ and finds $$x = \frac{5}{2}$$ and $$x = \frac{7}{2}. $$What mistake did the student make?

Accepted Answer

They added instead of subtracting for one solution.

Question 15

Which of the following is the correct solution for $$|x+2| = 2$$?

Accepted Answer

$$x = 0 or$$ $$x = -4$$

Question 16

If $$|x| \u003e a$$ for $$a \u003e 0$$, what is the equivalent compound inequality?

Accepted Answer

$$x  a$$

Absolute Value Equations and Inequalities: Concepts, Methods, and Applications

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