1. September 20, 2011 At the end of today, you will be able to Solve inequalities and compound inequalities Warm-up: Solve for x 1.│5x│ - 38 = -8 2. -6x + 8 = -14x – 28 HW 1.5: Pg. 37 #15-23odd, 33, Pg. 44 #8, 27

2. 1.5 Inequalities An inequality is a number sentence with more than one solution. For example: x > -4 “The solution of the inequality are all real #s greater than -4.” Graph: 20 -15-5515 -20 -1010-20200

3. Two ways to express your answer: Set Builder Notation “The set of all numbers x such that x is greater than 9” Interval Notation The infinity symbols +∞ and -∞ are used to indicate that a set is unbounded in the positive or negative direction. To show that the endpoint is not included in the set, a parenthesis is used. “ ( “ -- does not include the value “ [ “ -- includes the value 20 -15-5515 -20 -1010-20200

4. Set Builder to Interval Notation For example: 20 -15-5515 -20 -1010-20200

5. 1.5 Solving Inequalities Example 1: We solve equations like this: add 8 + 8 +8 4x = -8 divide 4 44 x = -2 4x – 8 = -16 Are the steps the same for this inequality? 4x – 8 ≥ -16 + 8 +8 4x ≥ -8 44 x ≥ -2 What would the graphs look like for both solutions? Write the solution in set builder and interval notation.

6. Solving Inequality Practice Solve and Graph. Write in set builder and interval notation. 1. 2.3 > n + 6 3. -4x + 8 ≤ -24 #1 rule for inequalities: If you multiply or divide by a negative on both sides, switch the sign! For #3, do you remember the “extra” step to correctly find the solutions?

7. Compound Inequalities Compound Inequalities are two inequalities combined into one set of solutions. For example: {x| x ≥ 4 or x < 0} Graph the solutions: 20 -15-5515 -20 -1010-20200 What are some solutions of the set? What are not solutions of the set?

8. Solving and graphing an “or” compound inequality Example 2: Solve: x + 4 ≤ -2 or -12x < 6 Step 1: Solve as two separate inequalities Step 2: Graph both inequalities on one graph. - 4 -4 10 -8-4048 -10 -6210-2-106 x ≤ -6 -12 x >or What are some solutions? What are not solutions? {x | x ≤ -6 or x > }

9. You try… Solve and graph the inequality. 4. m – 7 ≥ -3 or -2m + 1 ≥ 11 5. 3a + 8 < 23 or HW 1.5: Pg. 37 #15-23odd, 33, Pg. 44 #8, 27

10. Example 3 Solve and graph: - 9 ≤ 4y – 3 ≤ 13 Solving and graphing an “and” compound inequality Step 1: Solve as two separate inequalities + 3 - 6 ≤ 4y ≤ 16 444 ≤ y ≤ 4 Step 2: Graph both inequalities on one graph. 10 -8-4048 -10 -6210-2-106 { x | }

11. Practice… Solve each inequality and find the solutions set on a number line. 6. -14 ≤ 3x – 8 ≤ 16 7. 22 < 6w – 2 < 82 HW 1.5: Pg. 37 #19-23odd, 33, Pg. 44 #8, 9, 27, 28, 29