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# Solving Compound Inequalities Equations

Sep 17, 2022

## Key Concepts

• Define a compound inequality
• Solve a compound inequality involving “or”
• Solve a compound inequality involving “and”
• Use compound inequality to solve problems

### Compound Inequality

Step 1: Write an inequality to represent the solutions shown in each part of the graph.

Step 2: Draw the compound inequality.

The compound inequality describes the graph of x ≤ -3 or x > 2.

### Solve compound inequality including “or”

• When a compound inequality with an “or” is given, the graph shows all points that appear in either of the solutions above.

Example: Solve 2x-5 > 3 or -4x+7 < -25

Step 1: Solve and graph 2x-5 > 3

2x-5+5 > 3+5

2x > 8

2x/2 > 8/2

x > 4

Step 2: Solve and graph 4x-7 < 25

4x+7–7 < 25+7

4x < 32

4x/4 < 32/4

x < 8

The solutions can be 5, 6, 7.

### Solving compound inequality including “and”

• When a compound inequality with an “and” is given, the final graph shows all points that appear in both solutions above.

Example: Solve -12 ≤ 7x+9 < 16

Step 1: Solve and graph -12 ≤ 7x+9

-12 – 9 ≤ 7x + 9 – 9

-21 ≤ 7x

−217 ≤ 7x/7

-3 ≤ x

Step 2: Solve and graph 7x+9 < 16

7x+9-9 < 16-9

7x < 7

7x/7 < 7/7

x < 1

The solution is x-3 and  x<1, or -3≤x<1

## Exercise

• Solve 12<2x<28
• The compound function that represents the graph is _________.
• Write the compound inequality that represents the area A of the rectangle if 35≥A25
• Find the area of the right-angled triangle if the height is 5 units and the base is x units, given that the area of the triangle lies between 10 and 35 sq. units.
• Write an inequality that represents the quantity that is greater than 18 but less than or equal to 27

### What have we learned

• A compound equality is made up of two or more inequalities.

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