Question
Solve the inequality showing all of your steps.
|2x−7|≤13
Note: Enter your answer and SHOW ALL THE STEPS NECESSARY to solve this problem in the space provided. You will not receive credit without all work shown.
|2x−7|≤13
Note: Enter your answer and SHOW ALL THE STEPS NECESSARY to solve this problem in the space provided. You will not receive credit without all work shown.
Answers
Answered by
GPT-5 mini
We solve |2x − 7| ≤ 13.
1. Use the definition of absolute value inequality: |A| ≤ B (with B ≥ 0) is equivalent to −B ≤ A ≤ B.
Here A = 2x − 7 and B = 13, so
−13 ≤ 2x − 7 ≤ 13.
2. Add 7 to all three parts:
−13 + 7 ≤ 2x − 7 + 7 ≤ 13 + 7
−6 ≤ 2x ≤ 20.
3. Divide all parts by 2:
−6/2 ≤ x ≤ 20/2
−3 ≤ x ≤ 10.
4. Check endpoints:
x = −3: |2(−3) − 7| = |−13| = 13 ≤ 13 ✓
x = 10: |2(10) − 7| = |13| = 13 ≤ 13 ✓
Solution in inequality form: −3 ≤ x ≤ 10.
Solution in interval notation: [−3, 10].
1. Use the definition of absolute value inequality: |A| ≤ B (with B ≥ 0) is equivalent to −B ≤ A ≤ B.
Here A = 2x − 7 and B = 13, so
−13 ≤ 2x − 7 ≤ 13.
2. Add 7 to all three parts:
−13 + 7 ≤ 2x − 7 + 7 ≤ 13 + 7
−6 ≤ 2x ≤ 20.
3. Divide all parts by 2:
−6/2 ≤ x ≤ 20/2
−3 ≤ x ≤ 10.
4. Check endpoints:
x = −3: |2(−3) − 7| = |−13| = 13 ≤ 13 ✓
x = 10: |2(10) − 7| = |13| = 13 ≤ 13 ✓
Solution in inequality form: −3 ≤ x ≤ 10.
Solution in interval notation: [−3, 10].
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