1. 1.8 Solving Absolute Value Equations and Inequalities Objectives: Write, solve, and graph absolute value equations and inequalities in mathematical and real world situations. Standards: 2.8.11.A Analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically.

2. I. Absolute Value The absolute value of x is the distance from x to 0 on the number line.  -3  =  5  =  -256,000  = 3 5 256000

3. II. Absolute Value Equations – If a < 0 and  x  = a, then x = a or x = - a. Break up the original absolute value equation into 2 equations. One of the equations equals the positive value of the right side and the other equation equals the negative value.

4. Solve each absolute value equation. Graph the solution on a number line. ** Ex 1.  2x + 3  = 4 Ex 1.  2x + 3  = 4

5. Ex 1.  2x + 3  = 4

6. Solve each absolute value equation. Graph the solution on a number line. ** Ex 2.  3x + 5  = 7 Ex 2.  3x + 5  = 7

7. Solve each absolute value equation. Graph the solution on a number line. ** Ex 3.  x - 3  = 3x + 5 Ex 3.  x - 3  = 3x + 5 Check the solutions. Do they work? Graph the solution(s) on a number line.

8. Solve each absolute value equation. Graph the solution on a number line. ** Ex 4.  x – 4  = x + 1 Ex 4.  x – 4  = x + 1 Graph the solution(s) on a number line.

9. III. Absolute Value Inequalities If a > 0 and  x  -a. If a > 0 and  x  -a. If a > 0 and  x  > a, then x > a OR x 0 and  x  > a, then x > a OR x < -a. Break the original absolute value inequality into 2 inequalities. The first inequality is the same as the original problem, but it does not include the absolute value bars. For the second inequality, flip the inequality sign and change the sign of the number on the right side. GREATOR LESS THAND GREATOR LESS THAND

10. Solve the absolute value inequality. Graph the solution on the number line. Ex 1.  5 – 3x  < 9 Ex 1.  5 – 3x  < 9

11. Ex 1.  5 – 3x  < 9

12. Solve the absolute value inequality. Graph the solution on the number line. Ex 2.  5x – 3  < 7 Ex 2.  5x – 3  < 7 5x – 3 -7 5x < 10 x < 2 5x > -4 x > -4/5 Graph the solution. * -4/5 < x < 2

13. Solve the absolute value inequality. Graph the solution on the number line. Ex 3.  3x – 7  > 1 Ex 3.  3x – 7  > 1 3x – 7 > 1 or 3x – 7 < -1 3x > 8 x > 8/3 3x < 6 x < 2 Graph the solution. * x 8/3

14. Solve the absolute value inequality. Graph the solution on the number line. Ex 4.  5x + 2  > 8 Ex 4.  5x + 2  > 8 5x + 2 > 8 or 5x + 2 < -8 5x > 6 x > 6/5 5x < -10 x < -2 Graph the solution. * x 6/5

15. Writing Activities: Absolute-Value Equations 14). The right-hand side of the equation shown at right is an integer. Does the equation have 0, 1, or 2 solutions? Explain. 2x – 2 = ?