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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