College Algebra: Section 1 College Algebra: Section 1.6 Equations and Inequalities Involving Absolute Value Objectives of this Section Solve Equations Involving Absolute Value Solve Inequalities Involving Absolute Value

Equations Involving Absolute Value If the absolute value of an expression equals some positive number a, then the expression itself equals either a or -a. Thus,

Theorem

Ex. Solve y + 21 ≥ 7 y ≥ -14 (-14) + 21 ≥ 7 7 ≥ 7 ● - 21 -21 -15 -14 - 21 -21 y ≥ -14 (-14) + 21 ≥ 7 7 ≥ 7 Draw the “river” Subtract 21 from both sides Simplify Check your answer Graph the solution ● -15 -14 -13

Ex. Solve 8y + 3 > 9y - 14 - 8y - 8y 3 > y - 14 + 14 + 14 + 14 + 14 17 > y y < 17 8(16) + 3 > 9(16) – 14 131 > 130 Draw “the river” Subtract 8y from both sides Simplify Add 14 to both sides Rewrite inequality with the variable first Check your answer Graph the solution o 16 17 18

Ex. Solve 8y + 3 > 9y - 14 - 8y - 8y 3 > y - 14 + 14 + 14 + 14 + 14 17 > y y < 17 8(16) + 3 = 9(16) – 14 131 > 130 Big Tip!!! At the end of solving inequality, always put the variable at the LEFT hand side. Then arrow of the inequality sign tells you the correct graph y < 17 The graph should toward to the left. o 16 17 18

To Solve the Absolute Value Inequalities 1. Isolate the absolute value expression. 2. Make sure the absolute value inequality can be defined. 3. For any defined absolute value inequality with template: |X|  a, where a > 0 and  = <, >, ≤, ≥ write the corresponding compound inequalities by following the rules: a) if “|X| > a” or “|X| ≥ a”, meaning “greator”, “leaving the jail” b) set “jail boundaries” as “–a” and “a” c) write compound inequalities as X < –a or X > a meaning “stay left to the left boundary or right to the right boundary” d) if “|X| < a” or “|X| ≤ a”, meaning “less thand”, “going to the jail” e) set “jail boundaries” as “–a” and “a” f) write compound inequalities as –a < X < a (and) meaning “stay between the two boundaries” 4. Solve the converted compound inequalities.

–2 < x < 8 Absolute Value Inequality Solve | x – 3 | < 5 The absolute value expression is isolated It is a well defined absolute value inequality It is a “less thand” inequality. (go to the jail) set jail boundaries: –5, and 5 write compound inequalities: (stay in between the jail boundaries) –5 < x – 3 < 5 –2 < x < 8 +3 +3 +3

–5 < x < –3 Absolute Value Inequality You try this! Solve | x + 4 | < 1 The absolute value expression is isolated It is a well defined absolute value inequality It is a “less thand” inequality. (go to the jail) set jail boundaries: –1, and 1 write compound inequalities: (stay in between the jail boundaries) –1 < x + 4 < 1 –5 < x < –3 – 4 – 4 – 4

Absolute Value Inequality Solve | 2x + 3 | – 3 ≥ 2 The absolute value expression is NOT isolated | 2x + 3 | – 3 ≥ 2 | 2x + 3 | ≥ 5 It is a well defined absolute value inequality It is a “greator” inequality. (leaving the jail) set jail boundaries: –5, and 5 write compound inequalities: (stay left to left boundary and right to the right boundary) 2x + 3 ≤ –5 or 2x + 3 ≥ 5 2x ≤ –8 or 2x ≥ 2 x ≤ –4 or x ≥ 1 + 3 + 3 – 3 – 3 – 3 – 3

Absolute Value Inequality You try this! Solve | 4x – 3 | + 5 ≥ 8 The absolute value expression is NOT isolated | 4x – 3| + 5 ≥ 8 | 4x – 3 ≥ 3 It is a well defined absolute value inequality It is a “greator” inequality. (leaving the jail) set jail boundaries: –3, and 3 write compound inequalities: (stay left to left boundary and right to the right boundary) 4x – 3 ≤ –3 or 4x – 3 ≥ 3 4x ≤ 0 or 4x ≥ 6 x ≤ 0 or x ≥ 3/2 – 5 – 5 + 3 + 3 + 3 + 3

Summary Before solving absolute value equations or inequalities, you MUST isolate the absolute value expression and check them are defined or not. Remember telling yourself the jail story. It will help you set up the correct equations or compound inequalities. Solve the equations or compound inequalities and don’t forget the learned knowledge such as when multiplying or dividing a negative number, you need flip the inequality sign.

Solution Set:

Theorem