1. Chapter 12 Section 5 Solving Compound Inequalities

2. These two form a compound inequality. The compound inequality w>25 and w ≤ 50 can be written without using the word and. Method 1: 25< w ≤ 50 This can be read as 25 less that w, which is less than or equal to 50. Method 2: 50 ≥ w > 25 This can be read as 50 is greater than or equal to w, which is greater than 25. Both symbols are facing the same direction.

3. Example One Write x ≥ 2 and x < 7 as a compound inequality without using the word and. x ≥ 2 and x < 7 can be written as 2 ≤ x < 7 or as 7 > x ≥ 2

4. Your Turn Write x < 10 and x ≥ -4 as a compound inequality without using the word and. x < 10 and x ≥ -4 can be written as -4 ≤ x < 10 or as 10 > x ≥ -4

5. Your Turn Write x ≤ 6 and x ≥ 2 as a compound inequality without using the word and. x ≤ 6 and x ≥ 2 can be written as 2 ≤ x ≤ 6 or as 6 ≥ x ≥ 2

6. A compound inequality using and is true if and only if both inequalities are true. Thus, the graph of a compound inequality using and is the intersection of the graphs of the two inequalities.

7. Consider the inequality -2 -2 and x < 3. To graph, follow the steps.

8. Step 1: Graph x > -2 Step 2: Graph x < 3 Step 3: Find the intersection of the graphs. -5 -4 -3 -2 -1 0 1 2 3 4 5 The solution is {x l -2 < x < 3}

9. Example 2 Graph the solution 25 < w ≤ 50. Rewrite the compound inequality using and. 25 25 and w ≥ 50.

10. Step 1: Graph x > 25 Step 2: Graph x ≤ 50 Step 3: Find the intersection of the graphs. 10 15 20 25 30 35 40 45 50 55 60 The solution is {x l 25 < x ≤ 50} 10 15 20 25 30 35 40 45 50 55 60

11. Your Turn Graph the solution of 3 ≤ x ≤ 5 1 2 3 4 5 6 7 8

12. Example 3 Often, you must solve a compound inequality before graphing it. Solve 4 < x + 3 ≤ 12. Graph your solution. Step 1: Rewrite the compound inequality using and. 4 < x + 3 ≤ 12 x + 3 > 4 and x + 3 ≤ 12

13. Step 2 Solve each inequality. x + 3 > 4 x + 3 - 3 > 4 – 3 x > 1 x + 3 ≤ 12 x + 3 - 3 ≤ 12 - 3 x ≤ 9

14. Step 3 Rewrite the inequality as 1 < x ≤ 9. The solution in {x l 1 < x ≤ 9}. The graph of the solution is shown below. 0 1 2 3 4 5 6 7 8 9 10 11 12

15. Your Turn Solve -2 < x – 4 < 2. Graph the solution. 2 < x < 6 0 1 2 3 4 5 6 7 8 9 10 11 12

16. Another type of compound inequality uses the word or. This type of inequality is true if one or more of the inequalities is true. The graph of a compound inequality using or is the union of the graphs of the two inequalities.

17. Example 4 Graph the solution of x > 0 or x ≤ -1. Step 1: Graph x > 0 0 1 2 3 4 5 6 7 8 9 10 11 12 Step 2: Graph x ≤ -1 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 Step 3:Find the union of the graphs. -7 -6 -5 -4 -3 -2 -1 0 2 3 4 5 6

18. Your Turn Graph the solution of x > -3 or x < -5. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4

19. Sometimes you must solve compound inequalities containing the word or before you are able to graph the solution.

20. Example 5 Solve 3x ≥ 15 or -2x < 4. Graph the solution. 3x ≥ 15 3 x ≥ 5 Now graph the solution. -2x < 4 -2 x > -2

21. Example 5 Step 1: Graph x ≥ 5 0 1 2 3 4 5 6 7 8 9 10 11 12 Step 2: Graph x > -2 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 Step 3:Find the union of the graphs. -7 -6 -5 -4 -3 -2 -1 0 2 3 4 5 6 The last graph shows the solution{x l x > -2}.

22. Your Turn Solve -6x > 18 or x – 2 < 1. Graph the solution. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 {x l x < 3}