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Notes 2.5

Creating & Solving Compound Inequalities Solving compound inequalities involving “AND.” Combining two or more simple inequalities forms a compound inequality.

The graph of a compound inequality involving “AND” is the intersection of the two simple graphs.

𝑥 > 2 𝐴𝑁𝐷 𝑥 < 6

4 ≤ 𝑥 + 2 𝐴𝑁𝐷 𝑥 + 2 ≤ 8

4 ≤ 𝑥 + 2 𝐴𝑁𝐷 𝑥 + 2 ≤ 8

4 − 2 ≤ 𝑥 + 2 − 2 𝐴𝑁𝐷 𝑥 + 2 − 2 ≤ 8 − 2 2 ≤ 𝑥 𝐴𝑁𝐷 𝑥 ≤ 6

2 ≤ 𝑥 ≤ 6

– 5 ≤ 2𝑥 + 3 ≤ 9

– 5 ≤ 2𝑥 + 3 ≤ 9 – 5 ≤ 2𝑥 + 3 𝐴𝑁𝐷 2𝑥 + 3 ≤ 9

– 5 − 3 ≤ 2𝑥 + 3 − 3 𝐴𝑁𝐷 2𝑥 + 3 − 3 ≤ 8 − 3 – 8 ≤ 2𝑥 𝐴𝑁𝐷 2𝑥 ≤ 5 −8 2 ≤ 2𝑥 2

𝐴𝑁𝐷 2𝑥 2 ≤ 5 2

– 4 ≤ 𝑥 ≤ 2 1 2

Solving compound inequalities involving “OR.” Combining two or more simple inequalities forms a compound inequality. The graph of a compound inequality involving “OR” is the union of the two simple graphs.

𝑥 < 2 𝑂𝑅 𝑥 > 6

– 4 + 𝑥 > 1 𝑂𝑅 – 4 + 𝑥 < 3

– 4 + 𝑥 > 1 𝑂𝑅 – 4 + 𝑥 < 3

– 4 + 4 + 𝑥 > 1 + 4 𝑂𝑅 – 4 + 4 + 𝑥 < 3 + 4

𝑥 > 5 𝑂𝑅 𝑥 < 7

2𝑥 ≤ 6 𝑂𝑅 3𝑥 > 15

2𝑥 ≤ 6 𝑂𝑅 3𝑥 > 15 2𝑥 2 ≤ 6 2

𝑂𝑅 3𝑥 3 > 15 3

𝑥 ≤ 3 𝑂𝑅 𝑥 > 5

You try: −10 < 3𝑥 + 2 < 8

4𝑥 − 1 < 15 𝑂𝑅 8𝑥 ≥ 48