1. Compound Inequalities
Objective: To solve conjunctions and disjunctions

2. 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. 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

4. 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

5. 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

6. 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

7. 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

8. -2 < 3x + 1 or 3x < -9 x > -1 or x < -3
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9. 6 < 3x + 6 < 12 6 – 6 < 3x + 6 – 6 < 12 – 6
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10. 6 > 2x > -8 6/2 > 2x/2 > -8/2 3 > x > -4
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11. 2x – 1 > 3 or x – 2 < 3 2x – 1 + 1 > 3 + 1 2x > 45
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12. 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
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