Compound inequalities are statements with more than one inequality in each problem. Following is an example of a compound inequality.

 

-4 £ x < 2

 

The best way to look at this problem is by ignoring one part of it and focusing on the rest. The previous example is actually two inequalities, they are

4 £ x and x < 2.

 

 

The inequality on the left is read as “x is greater than or equal to – 4.” The inequality on the right is “x is less than 2.” By putting both of these together it can be read as “x is greater than or equal to 4 and x is less than 2.” The only numbers that work are the ones between –4 and 2. For example 1, it is larger than – 4 and it is less than 2, so 1 would be answer. The number 3 would not work. Even though 3 is larger than – 4, it is not smaller than 2.

 

Looking at another example may give a little more understanding.

 

2 < 2 x + 1 £ 5

 

Everything that is in the middle, 2 x +1, goes with both sides. So when breaking this one down, use 2 x + 1 for the left and the right sides.

 

2 < 2 x + 1 and 2 x + 1 £ 5

 

To solve a compound inequality, solve both sides of the previous inequality.

 

Example

Solve –7 £ 2 x –1 < 3

 

Separate the inequality.

–7 £ 2 x –1 and

2 x –1 < 3

 

Start with the first inequality and solve for x.

 

–7 £ 2 x –1

 

Add 1 to both sides and divide by 2.

 

–3 £ x

 

This means that x is greater than or equal to –3. Solve for x in the second inequality.

2 x –1 < 3

 

Add 1 to both sides and divide by 2.

 

x < 2 x is less than 2.

 

Rewrite the results from solving both of these into one inequality.

 

–3 £ x < 2

 

Answer! x is greater than or equal to –3 and less than 2.