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**Compound Inequalities**

Objective: To solve conjunctions and disjunctions

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Conjunction A sentence formed by joining two sentences with the word and. x > -3 and x < 4 in order for a value of x to make the statement true both conditions must be satisfied

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**Disjunction Formed by joining two sentences with the word or**

A solution only has to satisfy one of the conditions to be true x > 2 states that x > 2 or x = 2, notice that the dot is shaded in to show equality. x > 3 or x < 1 2 1 3

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**Solving Inequalities With Disjunction**

2x + 3 < 7 or -4x < -16 2x + 3 – 3 < 7 – 3 2x < 4 2x/2 < 4/2 x < 2 -4x < -16 -4x/-4 > -16/-4 x > 4 x < 2 or x > 4 2 4

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**Solving Inequalities With Conjunction**

4 < 2(x – 1) < 8 4 < 2(x – 1) and 2(x – 1) < 8 4 < 2(x – 1) 4 < 2x – 2 distribute 4 + 2 < 2x – 2 + 2 6 < 2x 6/2 < 2x/2 3 < x x > 3 2(x – 1) < 8 2x – 2 < 8 distribute 2x – < 8 + 2 2x < 10 2x/2 < 10/2 x < 5 x > 3 and x < 5 3 5

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**Alternative Solution For Conjunction**

4 < 2(x – 1) < 8 4 < 2x – 2 < 8 distribute 4 + 2 < 2x – < 8 + 2 6 < 2x < 10 6/2 < 2x/2 < 10/2 3 < x < 5 3 5

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**Try These! Solve and Graph each Compound Inequality**

-2 < 3x + 1 or 3x < -9 6 < 3x + 6 < 12 6 > 2x > -8 2x – 1 > 3 or x – 2 < 3 2x + 3 < 3 and x – 4 > 1 x > -1 or x < -3 Click for solution 0 < x < 2 -4 < x < 3 All Real Numbers Ø -3 -1 2 -4 3 End show

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**-2 < 3x + 1 or 3x < -9 x > -1 or x < -3**

Back to Try These!

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**6 < 3x + 6 < 12 6 – 6 < 3x + 6 – 6 < 12 – 6**

Back to Try These!

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**6 > 2x > -8 6/2 > 2x/2 > -8/2 3 > x > -4**

Back to Try These!

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**2x – 1 > 3 or x – 2 < 3 2x – 1 + 1 > 3 + 1 2x > 4**

5 Back to Try These!

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**2x + 3 < 3 and x – 4 > 1 2x + 3 – 3 < 3 – 3 2x < 0**

Because there is no value for x that satisfies x < 0 and x > 5 simultaneously there is no solution for the inequality Back to Try These!

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4.1 Solving Linear Inequalities

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