# SOLVING COMPOUND INEQUALITIES

## 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 :

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.

Example 2 :

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

Solution :

 -3h + 4 < 19Subtract 4 on both sides-3h + 4 - 4 < 19 - 4-3h < 15Divide by -3 on both sides,-3h/(-3) < 15/ (-3)h < -5 7h - 3 > 18.Add 3 on both sides,7h - 3 + 3 > 18 + 37h > 21Divide by 7 on both sides7h/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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