Compound Inequalities

Objective: To solve conjunctions and disjunctions

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.

-3 4

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 < 12

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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

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Solving Inequalities With Conjunction

• 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 – 2 + 2 < 8 + 2• 2x < 10• 2x/2 < 10/2• x < 5

4 < 2(x – 1) < 84 < 2(x – 1) and 2(x – 1) < 8

x > 3 and x < 5

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Alternative Solution For Conjunction

• 4 < 2(x – 1) < 8

• 4 < 2x – 2 < 8 distribute

• 4 + 2 < 2x – 2 + 2 < 8 + 2

• 6 < 2x < 10

• 6/2 < 2x/2 < 10/2

• 3 < x < 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• Click for solution• -4 < x < 3• Click for solution• All Real Numbers• Click for solution• Ø• Click for solution

-1-3

20

3-4

End show

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

• -2 – 1 < 3x + 1 – 1 • -3 < 3x • -3/3 < 3x/3• -1 < x• x > -1

• 3x/3 < -9/3• x < -3

• x > -1 or x < -3

Back to Try These!

6 < 3x + 6 < 12

• 6 – 6 < 3x + 6 – 6 < 12 – 6

• 0 < 3x < 6

• 0/3 < 3x/3 < 6/3

• 0 < x < 2

Back to Try These!

6 > 2x > -8

• 6/2 > 2x/2 > -8/2

• 3 > x > -4

• -4 < x < 3

Back to Try These!

2x – 1 > 3 or x – 2 < 3

• 2x – 1 + 1 > 3 + 1• 2x > 4• 2x/2 > 4/2• x > 2

• x – 2 + 2 < 3 + 2• x < 5

• x > 2 or x < 5

52

Back to Try These!

2x + 3 < 3 and x – 4 > 1

• 2x + 3 – 3 < 3 – 3 • 2x < 0• 2x/2 < 0/2• x < 0

• x – 4 + 4 > 1 + 4• x > 5

• 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!