Solve Absolute Value Equations Section 6-5 Solve Absolute Value Equations

Example 1 Solve an absolute value equation: | x | = 9 • The distance between x and 0 is 9. • There are always two answers for absolute value • So, x = 9 and x = -9

Solving an Absolute Value Equation The equation |ax + b| = c where c ≥ 0 is equivalent to the statement: ax + b = c OR ax + b = -c

Example 2 Solve an absolute value equation: |x + 4| = 3 x + 4 = 3 • Rewrite the absolute value equation as two equations. x + 4 = 3 x + 4 = -3 - 4 -4 - 4 -4 • Solve each equation separately. x = -1 x = -7

Example 3 Rewrite an absolute value equation: 3|4x + 2| - 7 = 11 + 7 +7 3|4x + 2| = 18 3 3 |4x + 2| = 6 •Rewrite the equation in the form |ax + b| = c. Need to get absolute value BY ITSELF FIRST! • Add 7 to both sides. •Divide both sides by 3. 4x + 2 = 6 4x + 2 = -6 • Solve the absolute value equation. - 2 -2 - 2 -2 4x = -8 4x = 4 4 4 4 4 x = 1 x = -2

Homework Section 6.5 Pg 393 - 395 4 – 6, 9, 14, 15, 16, 20, 24 – 26

Solve Absolute Value Inequalities Section 6-6 Solve Absolute Value Inequalities

Graphing Absolute Value Inequalities If the original problem has: > or ≥ Graphs goes AWAY from each other < or ≤ Graphs goes TOWARDS each other.

Example 1 Solve an absolute value inequality: | x | > 2 • There are always two answers for absolute value inequalities • Must make the number NEGATIVE and FLIP the symbol! So x > 2 and Graph: -8 -6 -4 -2 0 2 4 6 x < -2 Original problem was >, so graph goes AWAY!

Example 1 - Continued Solve absolute value inequalities: b) |x| ≤ 1.5 x ≤ 1.5 and Graph: -2 -1 0 1 2 All real numbers less than or equal to 1.5 and greater than or equal to -1.5. x ≥ -1.5 Original problem was <, so graph goes towardS!

Example 2 Solve an absolute value inequality: |x + 2| > 1 -2 -2 x > -1 Graph: -8 -6 -4 -2 0 2 4 6 Rewrite as a compound inequality. x + 2 < -1 Solve each inequality. -2 -2 x < -3 Original problem was >, so graph goes AWAY!

Example 3 Solve an absolute value inequality: |2x + 3| - 4 ≤ 5 + 4 +4 + 4 +4 |2x + 3| ≤ 9 2x + 3 ≤ 9 2x + 3 ≥ -9 - 3 -3 2x ≤ 6 2 2 x ≤ 3 Graph: -8 -6 -4 -2 0 2 4 6 •Isolate the absolute value. • Add 4 to both sides. •Rewrite as a compound inequality. •Solve each inequality. -3 -3 2x ≥ -12 2 2 x ≥ -6 Original problem was ≤, so graph goes TOWARDS!

Homework Section 6.6 Pg 401 - 403 6 – 13, 18