Chapter 12 Section 5 Solving Compound Inequalities

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.

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

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

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

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.

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

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

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

Step 1: Graph x > 25 Step 2: Graph x ≤ 50 Step 3: Find the intersection of the graphs The solution is {x l 25 < x ≤ 50}

Your Turn Graph the solution of 3 ≤ x ≤

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

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

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

Your Turn Solve -2 < x – 4 < 2. Graph the solution. 2 < x <

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.

Example 4 Graph the solution of x > 0 or x ≤ -1. Step 1: Graph x > Step 2: Graph x ≤ Step 3:Find the union of the graphs

Your Turn Graph the solution of x > -3 or x <

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

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

Example 5 Step 1: Graph x ≥ Step 2: Graph x > Step 3:Find the union of the graphs The last graph shows the solution{x l x > -2}.

Your Turn Solve -6x > 18 or x – 2 < 1. Graph the solution {x l x < 3}