**Solving compound inequalities:**

Here we are going to learn how to solve compound inequality.

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).

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.

**Example 1 :**

Graph the solution set of x < 3 and x ≥ -2

**Solution :**

Let us draw the graph for the first given inequality x < 3. In the number line we have to shade the portion which is lesser than 3.

Let us draw the graph for the first given inequality x ≥ -2. In the number line we have to shade the portion which is greater than or equal to -2.

The overlaps of the above two graphs is between -2 and 3.

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 2 :**

Solve -3h + 4 < 19 or 7h - 3 > 18. Then graph the solution set.

**Solution :**

-3h + 4 < 19 Subtract 4 on both sides -3h + 4 - 4 < 19 - 4 -3h < 15 Divide by -3 on both sides, -3h/(-3) < 15/ (-3) h < -5 |
7h - 3 > 18. Add 3 on both sides, 7h - 3 + 3 > 18 + 3 7h > 21 Divide by 7 on both sides 7h/7 > 21/7 h > 3 |

Let us draw the graph for the first given inequality h> 3. In the number line we have to shade the portion which is greater than 3.

Let us draw the graph for the first given inequality x < -5. In the number line we have to shade the portion which is lesser than -5.

By combining the above two graphs, we get

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