Math · College algebra · Worked example
Solve a compound inequality
Solve −1 < 5 − 2x ≤ 9 and write the answer in interval notation.
Work on all three parts at once
A three-part inequality is an “and” statement: 5 − 2x must be greater than −1 and at most 9. Whatever you do to the middle, do to both outer parts. Subtract 5 from all three.
Divide by −2 and reverse both signs
Dividing every part by −2 reverses both inequality signs: −6 ÷ (−2) = 3 and 4 ÷ (−2) = −2.
Rewrite from smallest to largest
Read the same statement from right to left so the smaller number comes first. The endpoint −2 is included, because of ≥, and 3 is left out.
Write the interval and check
The interval is [−2, 3). Check x = 0: 5 − 0 = 5, and −1 < 5 ≤ 9 is true. Check x = 3: 5 − 6 = −1, and −1 < −1 is false, so 3 stays out.
Result
−2 ≤ x < 3, which is [−2, 3).
Your turn
Solve 1 ≤ (3 − x)/2 < 4 and write the answer in interval notation.
Show the answer and explanation
−5 < x ≤ 1, which is (−5, 1].
Multiply all three parts by 2: 2 ≤ 3 − x < 8. Subtract 3: −1 ≤ −x < 5. Multiply by −1 and reverse both signs: 1 ≥ x > −5, which reads −5 < x ≤ 1.
Keep exploring
Open the steps in Math and leave the signs unreversed after dividing by −2: the checker marks that line ✗.
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