Solving Compound and Absolute Value Inequalities Chapter 1, Section 6

Don’t Forget: Solving inequalities follows the same procedures as solving equations. There are a few special things to consider with inequalities: We need to look carefully at the inequality sign. We also need to graph the solution set. If we multiply/divide by a negative number, flip the direction of the inequality.

How to graph the solutions > Graph any number greater than. . . open circle, line to the right < Graph any number less than. . . open circle, line to the left Graph any number greater than or equal to. . . closed circle, line to the right Graph any number less than or equal to. . . closed circle, line to the left

Solving Compound Inequalities Compound inequalities are 2 or more inequalities that are joined together by the words and, or “And” inequalities are called intersections and usually refer to less than problems “Or” inequalities are called unions and usually refer to greater than problems To solve compound inequalities, solve both parts of the inequality, graph both solutions, combine graphs according to “and”, “or”

What is the difference between and and or? AND means intersection -what do the two items have in common? OR means union -if it is in one item, it is in the solution A B

● ● ● 1) Graph x < 4 and x ≥ 2 a) Graph x < 4 o o b) Graph x ≥ 2 3 4 2 o 3 4 2 o b) Graph x ≥ 2 3 4 2 ● ● c) Combine the graphs d) Where do they intersect? ● 3 4 2 o

● ● 2) Graph x < 2 or x ≥ 4 a) Graph x < 2 o o b) Graph x ≥ 4 3 4 2 o 3 4 2 o b) Graph x ≥ 4 3 4 2 ● 3 4 2 ● c) Combine the graphs

Examples of graphing solutions: Examples of writing solutions:

Solve: 5k+2<-13 or 8k-1>19

Solve: 4d-1>-9 or 2d+5<11

Solve: y-2>-3 or y+4≥-3

Solve: 3a+8≤23 or a-6>7

Solving Absolute Value Inequalities Solving absolute value inequalities is a combination of solving absolute value equations and compound inequalities. Rewrite the absolute value inequality. For the first equation, all you have to do is drop the absolute value bars. For the second equation, you have to negate the right side of the inequality and reverse the inequality sign. Solve the TWO inequalities, graph your solution on a number line. Don’t forget to isolate the absolute value if needed!!

Solve: |2x + 4| > 12 Why is this problem an “or” problem??

In Conclusion Exit Slip: How would you explain the concept of “or” problems to a non-math student? Homework – 1.6 in workbook #’s 9, 12, 14, 15, 17, 20 (standard) #’s 6, 7, 9, 11, 14 (honors)

Recall: Compound inequalities use the words “and”, “or” “and” problems are called unions, and the solution involves everything that is in common. Absolute Value inequalities involve solving compound inequalities. Don’t forget to isolate the absolute value first!!

Solve: -10<3x+2 and 3x+2≤14

Solve: y-2>-3 and y+4≥-3

Solve: 13>2x+7 and 2x+7≥17

Solve: 18<4x-10<50

Solve: 2|4 - x| < 10 Why is this problem an “and” problem??

In Conclusion Exit Slip: Using 3-5 characteristics, compare/contrast intersections and unions Homework – 1.6 in workbook #’s 10, 11, 13, 16, 18, 19 (standard) #’s 5, 8, 10, 12, 13, 15, 16 (honors)