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

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

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

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

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

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

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

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

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

Writing $x < a$ instead of $-a < x < a$

Question 1

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

Accepted Answer

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

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 $|x + 1| < 5$?

Accepted Answer

$-6 < x < 4$

Question 6

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

Accepted Answer

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

Question 7

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

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

$|T - 98.6| < 2$

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

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

Question 16

Solve the inequality $|3x - 2| > 7$.

Accepted Answer

$x < -\dfrac{5}{3}$ or $x > 3$

Question 17

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

Accepted Answer

Writing $x < a$ instead of $-a < x < a$

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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