Absolute Value Inequalities

Show your work for each problem.

1.
Which compound inequality represents ?
1. a)
2. b)
3. c)
4. d) or
Answer:
means is less than 5 units from zero. This gives us (values between and ).
2.
Which compound inequality represents ?
1. a)
2. b) or
3. c)
4. d)
Answer: or
means is more than 4 units from zero. This gives us OR (values outside the interval).
3.
Solve . What is the lower bound of the solution? (Enter only the number)
Answer: -3
means . The lower bound is .
4.
Solve . What is the upper bound of the solution?
Answer: 7
means . The upper bound is .
5.
Solve
Answer: -1 < x < 7
1. What type of inequality is this (less than or greater than)? less than
2. Write the compound inequality: 4
3. Add 3 to all parts. What is the lower bound? -1
4. What is the upper bound? 7
6.
Solve
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 -5
3. Solve . What is ? x < -7
4. The second inequality is . What is ? x > 3
7.
What is the solution to ?
1. a)
2. b)
3. c)
4. d) or
Answer:
means . Adding 1: .
8.
Solve . What is the upper bound of the solution?
Answer: 5
means . Dividing by 2: . Upper bound is .
9.
What is the solution to ?
1. a) or
2. b)
3. c) or
4. d)
Answer: or
gives OR . Solving: or .
10.
Solve
Answer: -2 <= x <= 6
1. Write as compound inequality: 12
2. Add 6 to all parts: 18
3. Divide by 3. What is the lower bound? -2
4. What is the upper bound? 6
11.
What is the solution to ?
1. a) or
2. b)
3. c)No solution
4. d)All real numbers
Answer: No solution
Absolute value is always . Since for all , it can never be less than . No solution.
12.
What is the solution to ?
1. a)
2. b) or
3. c)No solution
4. d)All real numbers
Answer: All real numbers
Since and , the inequality is always true. Solution: all real numbers.
13.
Solve
Answer: x < -5 or x > 1
1. This is a greater than inequality. Write the first case: -12
2. Solve . First subtract 8: -20
3. Divide by 4: -5
4. Now solve . Subtract 8, then divide by 4: 1
14.
A machine part must be 50 mm with tolerance 0.5 mm. The inequality is . What is the maximum acceptable diameter?
Answer: 50.5
means . Adding 50: . Maximum is mm.
15.
Solve
Answer: x <= -2 or x >= 5
1. This is greater than or equal. First case: -7
2. Add 3: -4
3. Divide by 2: -2
4. Second case: solve . What is ? x >= 5
16.
For the inequality , what is the boundary value that separates the two solution regions?
Answer: 6
The boundary value is . Solutions are or , so values beyond 6 units from zero.
17.
Solve
Answer: -6 <= x <= -2
1. Write as compound inequality: 2
2. Subtract 4 from all parts. Lower bound: -6
3. Upper bound: -2