Absolute Value Inequalities

An absolute value inequality is

• an inequality that contains an absolute value expression
• impossible to solve
• an equation that contains an absolute value expression
• always going to have two solutions

| x + 3 | ≤ 2

• No Solution
• All Real Numbers
• -5 ≤ x ≤ -1
• 1 ≤ x ≤ 5

| x | ≤ 3.5

• No Solution
• All Real Numbers
• -3.5 ≤ x ≤ 3.5
• x ≤ -3.5  or  x ≥ 3.5

| 3c - 5 | < -2

• No Solution
• All Real Numbers
• 1 < c < ⁷/₃
• c < 1  or  c > ⁷/₃

| x + 1 | ≥ 3

• No Solution
• All Real Numbers
• x ≤ -4  or  x ≥ 2
• -4 ≤ x ≤ 2

| x + 3 | > 8

• No Solution
• All Real Numbers
• x < -11  or  x > 5
• -5 < x < 11

3 | 2t - 3 | - 5 > 16

• No Solution
• All Real Numbers
• t < -2  or  t > 5
• -2 < t < 5

What is the correct first step?

• divide by 10
• multiply by 10
• make it two inequalities with an "or"
• make it two inequalities with and "and

| -8x | < 8

What is the correct first step?

• separate into two inequalities with an "and"

• separate into two inequalities with an "or"

• divide both sides by -8

• add 8 to both sides

BEFORE solving an absolute value inequality, one can tell how the graph should appear based on the symbol. Which symbols give an "AND" graph solution?

• Less Than / Less Than or Equal To

• Greater Than / Greater Than or Equal To