Question 1

Which of the following is the correct property for solving absolute value inequalities of the form $$|X| \geq a$$, where $$a is a $$positive number?

Accepted Answer

$$X \leq -a or$$ $$X \geq a$$

Question 2

Solve the inequality $$\left| \dfrac{5x + 5}{2} \right| \geq 5$$ and express the solution in interval notation.

Accepted Answer

$$(-\infty, -3] \cup [1, \infty)$$

Question 3

What is the first step when solving $$\left| \dfrac{5x + 5}{2} \right| \geq 5$$?

Accepted Answer

Rewrite as two inequalities: $$\dfrac{5x + 5}{2} \leq -5 or$$ $$\dfrac{5x + 5}{2} \geq 5$$

Question 4

After clearing fractions in the inequality $$\dfrac{5x + 5}{2} \leq -5$$, what is the resulting equation?

Accepted Answer

$$5x + 5 \leq -10$$

Question 5

A medical researcher is analyzing the range of safe dosages for a drug, modeled by $$|d - 50| \geq 10$$, where $$d is $$the dosage in mg. What is the set of safe dosages?

Accepted Answer

$$(-\infty, 40] \cup [60, \infty)$$

Question 6

Which of the following best describes the solution set for $$|x| \geq a$$ where $$a \u003e 0$$?

Accepted Answer

All real numbers except $$(-a, a)$$

Question 7

If $$|x - 2| \geq 7$$, what is the solution set for $$x$$?

Accepted Answer

$$(-\infty, -5] \cup [9, \infty)$$

Question 8

A company requires that the temperature in its warehouse stays at least 8 degrees away from 20°C for safety. Which inequality models this requirement?

Accepted Answer

$$|T - 20| \geq 8$$

Question 9

Solve for $$x$$: $$|2x - 4| \geq 6$$

Accepted Answer

$$(-\infty, -1] \cup [5, \infty)$$

Question 10

Which step is necessary to solve $$|ax + b| \geq c$$ for $$x$$, where $$a$$, $$b$$, and $$c$$ are constants and $$c \u003e 0$$?

Accepted Answer

Rewrite as $$ax + b \leq -c or$$ $$ax + b \geq c$$

Question 11

A chemical reaction is safe if the pH is at least 2 units away from neutral (pH 7). Which interval represents the safe pH values?

Accepted Answer

$$(-\infty, 5] \cup [9, \infty)$$

Question 12

If $$|x + 3| \geq 4$$, what is the solution set for $$x$$?

Accepted Answer

$$(-\infty, -7] \cup [1, \infty)$$

Question 13

Which of the following is NOT a correct step when solving $$|x| \geq 5$$?

Accepted Answer

Rewrite as $$x \leq 5 or$$ $$x \geq -5$$

Question 14

A student claims that the solution to $$|x| \geq 0 is$$ $$x = 0. Is $$this correct?

Accepted Answer

No, the solution is all real numbers

Question 15

If $$|x - 10| \geq 3$$, which values of $$x$$ satisfy the inequality?

Accepted Answer

$$(-\infty, 7] \cup [13, \infty)$$

Solving Absolute Value Inequalities (College Algebra)

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