Solving Absolute Value Equations Solving Compound/Absolute Value Inequalities

1. Solving Absolute Value EquationsSolving Compound/Absolute Value Inequalities Lessons 1-4 and 1-6

2. Content Objectives • Solve Absolute Value Equations • Apply definition to solve, including ‘2’ possible equations • Recognize when there is no solution • Solve Compound Inequalities • Recognize unions versus intersections • Graph solutions/write solutions in Interval Notation • Solve Absolute Value Inequalities • Recognize unions versus intersections • Recognize no solution and infinite solutions situations

3. Solving Absolute Value Equations • Make the AV expression into a BOX (connect 2 vertical notations). • Solve the equation for the BOX. Make sure the coefficient is 1 and there are other terms with the BOX. Nothing gets into or out of the box!!!!! • Make sure your relationship is a true one . . . that is, make sure the BOX is equal to zero or a positive value. Otherwise the statement is FALSE (no solution) and you are finished. (why)

4. Solving AV Equations, Contd. • If the statement is TRUE, branch your equation, from the BOX (break open the box, the vertical lines will be GONE), one where the expression equals the given value, one where the expression equals the OPPOSITE of the given value. • Solve the two equations . • Check each solution to make sure true statements are generated by the found solutions. If a ‘solution’ generates a false statement, it is called extraneous, and you do NOT include it as part of your final solution.

5. Practice: Solving AV Equations |x + 15 | = 37 {-52, 22} |x - 4 | - 5 = 0 {-1, 9} |5x + 9 | + 16 = 2 {no solution} 5x + 24 = |8 – 3x | {-2} 40 – 4x = 2|3x + 10 | {-10, 6} |1/3x + 3| = -1 {no solution} ½ |6 – 2x | = 3x + 1 {1/2}

6. Solving Compound Inequalities • Compound Inequalities are composed of two inequalities joined by the words and(cookie) or or. • And implies a JOINT solution (intersection). BOTH inequalities must be satisfied by the solutions. • Or implies EITHER expression must be true with the solutions offered . . . called a union

7. Solving Compound Inequalities, Contd. • Solve each individual inequality. Make sure you follow the rules for solving inequalities. • Hint: if you have a cookie, break it into each individual inequality and make sure you recognize this as an AND (intersection) inequality. • Indicate the critical values on a number line (the two values you found when solving). • Underneath the number line sketch your inequality arrows.

8. Solving Compound Inequalities, Contd. • Evaluate which sections of the number line apply as part of your solution. • For unions, anywhere there is a TRUTH is included in the solution • For intersections, any section where BOTH STATEMENTS ARE TRUE are included in the solution set. • It is possible to have no solution . . . when you have an intersection and no where on the number line are both inequalities true. • It is possible to have infinite solutions (TFARN) . . . when you have a union and every section on the number line contains a TRUTH. (watch out for pure inequalities at the critical points)

9. Practice: Solving Compound Inequalities {c | c < 0 or c > 2} {y | -2 ≤ y ≤ 1} {x | -2 < x < -1} {a | a < -3 or a ≥ -2} {x | c < 2 < x ≤ 7} {w| w єR} (TFARN)

10. Solving Absolute Value Inequalities • AV Inequalities are compound inequalities in disguise. • Never, ever forget the rules for AV . . . the BOXED value must be equal to or greater than zero! Watch out for NO SOLUTIONS!! (so much to remember!) • Any AV Inequality that is less than/equal to a value is an INTERSECTION. Solutions will be included where BOTH sections of the number line include truths. • Any AV Inequality that is greater than/equal to a value is an UNION. Solutions will be located where any section of the number line contains a truth.

11. Practice: Solving AV Inequalities {t | t ≤ -3 or t ≥ 3 } {x | -2 < x < 2} {x | -2 ≤ x ≤ 0} {no solution} {x| xєR} (TFARN) {h | h ≤ -6 or h ≥ 4 }

12. Assignment • Page 31: 35 – 456 • Page 44: 29 - 42