**Solving compound inequalities examples :**

Here we are going to see some examples based on the concept solving compound inequalities.

A compound inequality is an equation with two or more inequalities joined together with either "and" or "or" (for example, x ≥ 6 and x < 3; x < -12 or x ≥ 8).

**Compound inequalities with "And" : **

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

In other words, the solution must be a solution of both inequalities.

The intersection can be found by graphing each inequality and then determining where the graphs overlap.

**Compound inequalities with "Or" : **

Another type of compound inequality contains the word or. A compound inequality containing or is true if one or more of the inequalities is true. The graph of a compound inequality containing or is the of the graphs of the two inequalities.

In other words, the solution of the compound inequality is a solution of either inequality, not necessarily both. The union can be found by graphing each inequality.

**Example 1 :**

Solve the compound inequality. Then graph the solution set

k + 2 > 12 and k + 2 ≤ 18

**Solution :**

k + 2 > 12 Subtract by 2 on both sides k + 2 - 2 > 12 - 2 k > 10 |
k + 2 ≤ 18 Subtract by 2 on both sides k + 2 - 2 ≤ 18 - 2 k ≤ 16 |

By graphing the inequality k > 10, we get the graph given below.

By graphing the inequality ≤ 16, we get the graph given below.

By combining the above two graphs, we get the common region between 10 and 16.

**Example 2 :**

Solve the following inequality and graph the solution

3 < 2*x -* 3 < 15

**Solution
:**

3
< 2*x -* 3 < 15

Add 3 on both sides

3 + 3 < 2x – 3 + 3 < 15 + 3

6 < 2x < 18

Divide by 2

6/2 < 2x/2 < 18/2

3 < x < 9

**Example 3 :**

Solve the following inequality and graph the solution

3*t
–* 7 ≥ 5 and 2*t
+* 6 ≤ 12

**Solution : **

3*t –* 7 ≥ 5 and 2*t +* 6 ≤ 12

3 Add 7 on both sides 3t – 7 + 7 ≥ 5 + 7 3t ≥ 12 Divide by 3 on both sides 3t/3 ≥ 12/3 t ≥ 4 |
2t + 6 ≤ 12 Subtract 6 on both sides 2t + 6 – 6 ≤ 12 – 6 2t ≤ 6 Divide by 2 on both sides t ≤ 3 |

t ≥ 4 and t ≤ 3

**Example 4 :**

Solve the following inequality and graph the solution

4m - 5 > 7 or 4m - 5 < -9

**Solution :**

4m - 5 > 7 Add 5 on both sides 4m - 5 + 5 > 7 + 5 4m > 12 Divide by 4 on both sides 4m/4 > 12/4 m > 3 |
4m - 5 < -9 Add 5 on both sides 4m - 5 + 5 < -9 + 5 4m < -4 Divide by 4 on both sides 4m/4 < -4/4 m < -1 |

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