Solving and Graphing Absolute Value Equations and Inequalities in One Variable

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

| k - 3 | < -1

• No Solution
• All Real Numbers
• 2 < k < 4
• k < 2  or  k > 4

| 2w - 1 | < 11

• No Solution
• All Real Numbers
• -5 < w < 6
• -6 < w < 5

| 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

3 | d + 1 | - 7 ≥ -1

• No Solution
• All Real Numbers
• d ≤ -3  or  d ≥ 1
• -1 < d < 3