Absolute Value Problems

1. Absolute Value Problems • Absolute value can be thought of as the distance a number is from zero • The graphs below show the solution set for each open sentence • |x| = 1 x = -1 or x = 1The distance between x and 0 is equal to 1 • |x| > 1 x < -1 or x > 1The distance between x and 0 is greater than 1 • |x| < 1 -1 < x < 1The distance between x and 0 is less than 1

2. Absolute Value Problems • An absolute value problem can be solved by changing it to an equivalent compound equality or inequality • |x| = 1 is equivalent to the compound equality x = -1 or x = 1 • |x| > 1 is equivalent to the compound “or” inequality x < -1 or x > 1 • |x| < 1 is equivalent to the compound “and” inequality -1 < x < 1 • Solve each absolute value problem by first changing it to an equivalent compound equality or inequality • |x – 2| = 8 becomes x – 2 = -8 or x – 2 = 8 so x = -6 or x = 10 • |3x – 2| ≥ 4 becomes 3x – 2 ≤ -4 or 3x – 2 ≥ 4 so x≤ -⅔ or x≥ 2 • |4x + 3| < 5 becomes -5 < 4x + 3 < 5 so -2 < x < ½

3. Absolute Value Problems • The absolute value expression must be isolated before the open sentence can be rewritten • For example, to solve |2x – 1| + 3 ≥ 8 • Subtract 3 from both sides to get |2x – 1| ≥ 5 • Rewrite this as a compound “or” inequality 2x – 1 ≤ -5 or 2x – 1 ≥ 5 • Solve the compound “or” inequality to get x≤ -2 or x≥ 3 • It is possible for absolute value problems to have solutions that are all real numbers or the empty set • For example: • |x + 4| > -1 is true for all real numbers • |x + 4| < -1 has no solution