Absolute Value Inequalities

Show your work for each problem.

1.
Which compound inequality represents $|x| < 5$ ?
1. a) $-5 < x < 5$
2. b) $x > -5$
3. c) $x < 5$
4. d) $x < -5$ or $x > 5$
Answer: $-5 < x < 5$
$|x| < 5$ means $x$ is less than 5 units from zero. This gives us $-5 < x < 5$ (values between $-5$ and $5$ ).
2.
Which compound inequality represents $|x| > 4$ ?
1. a) $x > 4$
2. b) $-4 < x < 4$
3. c) $x < -4$ or $x > 4$
4. d) $x < -4$
Answer: $x < -4$ or $x > 4$
$|x| > 4$ means $x$ is more than 4 units from zero. This gives us $x < -4$ OR $x > 4$ (values outside the interval).
3.
Solve $|x| < 3$ . What is the lower bound of the solution? (Enter only the number)
Answer: -3
$|x| < 3$ means $-3 < x < 3$ . The lower bound is $-3$ .
4.
Solve $|x| \leq 7$ . What is the upper bound of the solution?
Answer: 7
$|x| \leq 7$ means $-7 \leq x \leq 7$ . The upper bound is $7$ .
5.
Solve $|x - 3| < 4$
Answer: $-1 < x < 7$
1. What type of inequality is this (less than or greater than)? less than
2. Write the compound inequality: $-4 < x - 3 < ?$ 4
3. Add 3 to all parts. What is the lower bound? -1
4. What is the upper bound? 7
6.
Solve $|x + 2| > 5$
Answer: $x < -7$ or $x > 3$
1. What type of inequality is this? greater than
2. For greater than, we write two separate inequalities. The first is $x + 2 < ?$ -5
3. Solve $x + 2 < -5$ : $x < ?$ -7
4. The second inequality is $x + 2 > 5$ . Solve it: $x > ?$ 3
7.
What is the solution to $|x - 1| \leq 6$ ?
1. a) $-5 \leq x \leq 7$
2. b) $-6 \leq x \leq 6$
3. c) $-7 \leq x \leq 5$
4. d) $x \leq -5$ or $x \geq 7$
Answer: $-5 \leq x \leq 7$
$|x - 1| \leq 6$ means $-6 \leq x - 1 \leq 6$ . Adding 1: $-5 \leq x \leq 7$ .
8.
Solve $|2x| < 10$ . What is the upper bound of the solution?
Answer: 5
$|2x| < 10$ means $-10 < 2x < 10$ . Dividing by 2: $-5 < x < 5$ . Upper bound is $5$ .
9.
What is the solution to $|x + 5| \geq 3$ ?
1. a) $x \leq -8$ or $x \geq -2$
2. b) $-2 \leq x \leq 8$
3. c) $x \leq -2$ or $x \geq 8$
4. d) $-8 \leq x \leq -2$
Answer: $x \leq -8$ or $x \geq -2$
$|x + 5| \geq 3$ gives $x + 5 \leq -3$ OR $x + 5 \geq 3$ . Solving: $x \leq -8$ or $x \geq -2$ .
10.
Solve $|3x - 6| \leq 12$
Answer: $-2 \leq x \leq 6$
1. Write as compound inequality: $-12 \leq 3x - 6 \leq ?$ 12
2. Add 6 to all parts: $-6 \leq 3x \leq ?$ 18
3. Divide by 3. What is the lower bound? -2
4. What is the upper bound? 6
11.
What is the solution to $|x + 3| < -1$ ?
1. a) $x < -4$ or $x > -2$
2. b) $-4 < x < -2$
3. c)All real numbers
4. d)No solution
Answer: No solution
Absolute value is always $\geq 0$ . Since $|x + 3| \geq 0$ for all $x$ , it can never be less than $-1$ . No solution.
12.
What is the solution to $|2x - 5| > -3$ ?
1. a) $1 < x < 4$
2. b)All real numbers
3. c)No solution
4. d) $x < 1$ or $x > 4$
Answer: All real numbers
Since $|2x - 5| \geq 0$ and $0 > -3$ , the inequality is always true. Solution: all real numbers.
13.
Solve $|4x + 8| > 12$
Answer: $x < -5$ or $x > 1$
1. This is a greater than inequality. Write the first case: $4x + 8 < ?$ -12
2. Solve $4x + 8 < -12$ . First subtract 8: $4x < ?$ -20
3. Divide by 4: $x < ?$ -5
4. Now solve $4x + 8 > 12$ . Subtract 8, then divide by 4: $x > ?$ 1
14.
A machine part must have a diameter of 50 mm, with a tolerance of 0.5 mm: $|d - 50| \leq 0.5$ . What is the largest acceptable diameter $d$ in mm?
Answer: 50.5
$|d - 50| \leq 0.5$ means $-0.5 \leq d - 50 \leq 0.5$ . Adding 50: $49.5 \leq d \leq 50.5$ . Maximum is $50.5$ mm.
15.
Solve $|2x - 3| \geq 7$
Answer: $x ≤ -2$ or $x ≥ 5$
1. This is greater than or equal. First case: $2x - 3 \leq ?$ -7
2. Add 3: $2x \leq ?$ -4
3. Divide by 2: $x \leq ?$ -2
4. Second case: solve $2x - 3 \geq 7$ : $x \geq ?$ 5
16.
Solve $|x| > 6$ . The solution has two parts: $x < ?$ or $x > 6$ . What number goes in place of the question mark?
Answer: -6
$|x| > 6$ means $x$ is more than 6 units from zero: $x < -6$ or $x > 6$ . The missing number is $-6$ .
17.
Solve $|x + 4| \leq 2$
Answer: $-6 \leq x \leq -2$
1. Write as compound inequality: $-2 \leq x + 4 \leq ?$ 2
2. Subtract 4 from all parts. Lower bound: $-2 - 4 = ?$ -6
3. Upper bound: $2 - 4 = ?$ -2