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Compound Inequalities

A compound inequality (or combined inequality ) is two or more inequalities joined together with or or and .

To be a solution of an or inequality, a value has to make only one part of the inequality true.  To be a solution of an and inequality, it must make both parts true.

For example:

x 3 or x > 2

When two inequalities are joined with and , they are often written simply as a double inequality, like:

1 x < 2

(In other words, x 1 and x < 2 .)

Solving Compound Inequalities

Consider the compound inequality 2 < x + 3 < 7 .

To solve it, we need to subtract 3 -- not "from both sides", as you would do in a normal inequality, but from all THREE parts of the compound inequality.

2 3 < x < 7 3 5 < x < 4

In case you need to solve an "or" inequality, you can just treat the two inequalities separately.

3 x + 1 < 5 OR 2 x + 2 < 10

To solve the left part, first subtract 1 from both sides.

3 x < 6

Then divide both sides by 3 . Remember to reverse the inequality.

x > 2

For the second part, subtract 2 from both sides.

2 x < 12

Then divide both sides by 2 .

x < 6

So, the solution of the compound inequality is

x < 6 OR x > 2