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# Module 9

### Flash cards for studying Module 9 (Absolute Value Equations and Inequalities)

| x |= a If a is a positive number, | x |= a is equivalent to x = a, or x = -a. This can still be applied if the equation in the absolute value brackets is more complicated, such as | x + 10|= 5.
| x |= 5 Solve | x |= 5. | x |= 5 is equivalent to x = 5 and x = -5. Substitute each into the original equation. | x |= 5 | x |= 5 | 5 |= 5 | -5 |= 5 5 = 5; True 5 = 5; True Solution set: {5, -5}
4| x | + 50 = 46 4| x |+ 50 = 46 4| x |= -4 (Subtract 50) | x |= -1 (Divide by 4) Since the answer is negative, there is no solution.
| 3x-5 |= | 1-3x | 3x-5 = 1-3x 3x-5 = -(1-3x) 6x-5 = 1 3x = 4-3x 6x = 6 0 = 4 x = 1 Solution set is {1} since 0=4 is a false statement.
| 6+3a |-20 = -2 | 6+3a |-20 = -2 6+3a = 18 6+3a = -18 3a = 12 3a = 24 a = 4 a = -8 Solution set is {-8, 4}
| x | < a If a is positive, | x | < a is equivalent to -a < | x | < a. This also applies to the sign ≤
| x | ≤ 4 Refers to all numbers whose distance from 0 is less than or equal to 4. Solution set is [-4, 4]
| x-5 | < 7 | x-5 | < 7 x-5 < 7 x-5 > 7 x < 12 x > -2 Solution set is (-2, 12)
| 6x+3 |+3 < 24 | 6x+3 |+3 < 24 -21 < 6x + 3 < 21 (Subtract 3 from both sides and use the formula -a < | x | < a to set this inequality up.) -24 < 6x < 18 (Subtract 3 to get x alone.) -4 < x <3 (Divide by 6.) Solution is (-4, 3)
15 + | x | ≤ 8 15 + | x | ≤ 8 x ≤ -7 The inequality states that the answer should be less than or equal to 7 (less than 0, which cannot be the answer of an absolute value).
Created by: Nichollette