Solving Compound Inequalities

After studying this lesson, you will be able to:

• Solve compound inequalities.

Compound inequalities are two inequalities considered together.

A compound inequality containing the word and is true only if both inequalities are true. This type of compound inequality is called a conjunction.

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Examples of conjunctions:

x > -5 and x <1

y < 3 and y > -3

A compound inequality containing the word or is true if either of the inequalities is true. This type of compound inequality is called a disjunction.

Examples of disjunctions:

x > -5 or x > 1

y < 3 or y > -3

Example 1

Therefore, our answer is y> -3 or y <3

(this means that y can be any number since all numbers are either greater than -3 or less than positive 3)

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Example 2

Therefore, our answer is x < 3 and x > -3 (this means that x must be some number between 3 and -3 )

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Example 3

Therefore, our answer is m < 2 and m> 1

(this means that x must be some number between 1 and 2 )

Example 4

2 < 3x + 2 is the first inequality

3x + 2 < 14 is the second inequality

Now, we solve the way we did in Examples 1 -3: 2 < 3x + 2 and 3x + 2 < 14

Therefore, our answer is x> 0 and x <4 or we can write it 0 < x < 4

(this means that x must be some number between 0 and 4 )