Q: Which of the following is the correct piecewise definition for the absolute value function $$f(x) = |x|$$?

$$f(x) = \begin{cases} x & x \geq 0 \\ -x & x \u003c 0 \end{cases}$$

Q: If $$f(x) = |x - 3|$$, which of the following is the correct piecewise definition?

$$f(x) = \begin{cases} x - 3 & x \geq 3 \\ 3 - x & x \u003c 3 \end{cases}$$

Q: Which of the following is the correct piecewise definition for $$f(x) = |x + 3| - 2$$?

$$f(x) = \begin{cases} x + 1 & x \geq -3 \\ -x - 5 & x \u003c -3 \end{cases}$$

Question 1

Which of the following is the correct piecewise definition for the absolute value function $$f(x) = |x|$$?

Accepted Answer

$$f(x) = \begin{cases} x & x \geq 0 \ -x & x \u003c 0 \end{cases}$$

Question 2

Solve the absolute value equation $$|2x - 5| = 7. $$What are the solutions?

Accepted Answer

$$x = 1$$ and $$x = 6$$

Question 3

Which of the following graphs represents a vertical shift of the absolute value function $$f(x) = |x|$$ down by 3 units?

Accepted Answer

$$f(x) = |x| - 3$$

Question 4

Given the function $$f(x) = |x - 2| - 4$$, what is the vertex of its graph?

Accepted Answer

$$(2, -4)$$

Question 5

Which of the following is the correct solution to the inequality $$|x + 1| \u003c 5$$?

Accepted Answer

$$-6 \u003c x \u003c 4$$

Question 6

If $$f(x) = |x - 3|$$, which of the following is the correct piecewise definition?

Accepted Answer

$$f(x) = \begin{cases} x - 3 & x \geq 3 \ 3 - x & x \u003c 3 \end{cases}$$

Question 7

Solve the inequality $$-2|x| + 6 \leq 10. $$What is the solution set?

Accepted Answer

Question 8

Which of the following is the correct solution to $$|2x + 1| \geq 7$$?

Accepted Answer

$$x \leq -4 or$$ $$x \geq 3$$

Question 9

A medical thermometer is designed to alert if the temperature deviates by more than 2 degrees from 98.6°F. Which absolute value inequality models the safe temperature range?

Accepted Answer

$$|T - 98.6| \u003c 2$$

Question 10

Given $$f(x) = -|x - 2| + 5$$, which transformation has occurred to the parent function $$f(x) = |x|$$?

Accepted Answer

Shift right 2 units, reflect over x-axis, shift up 5 units

Question 11

Which of the following is the correct solution to $$|x - 4| = 0$$?

Accepted Answer

$$x = 4$$

Question 12

Solve the system of equations: $$\begin{cases} x + y = 5 \ x - y = 1 \end{cases}$$

Accepted Answer

$$x = 3, y = 2$$

Question 13

Which of the following is the correct solution to $$|x + 2| \leq 4$$?

Accepted Answer

$$-6 \leq x \leq 2$$

Question 14

A chemical process requires the concentration $$C to be $$within 0.5 units of 7.2. Which absolute value inequality models the acceptable concentration range?

Accepted Answer

$$|C - 7.2| \leq 0.5$$

Question 15

Which of the following is the correct piecewise definition for $$f(x) = |x + 3| - 2$$?

Accepted Answer

$$f(x) = \begin{cases} x + 1 & x \geq -3 \ -x - 5 & x \u003c -3 \end{cases}$$

Question 16

Solve the inequality $$|3x - 2| \u003e 7.$$

Accepted Answer

$$x  3$$

Question 17

Which of the following is a common mistake when solving $$|x|  0$$?

Accepted Answer

Writing $$x \u003c a$$ instead of $$-a \u003c x \u003c a$$

Question 18

Given $$f(x) = |x - 2| + 3$$, what is the minimum value of $$f(x)$$?

Accepted Answer

$$3$$

B6, B7

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