Numerical answer
How many integers x satisfy |2x − 7| < 9?
Show answer and solution
Answer: 8
|2x − 7| < 9 means −9 < 2x − 7 < 9.
Add 7 throughout: −2 < 2x < 16.
Divide by 2: −1 < x < 8.
The integers strictly between −1 and 8 are 0, 1, 2, 3, 4, 5, 6, 7 — that is 8 integers.
Tip: For |ax − b| < c (c > 0), the solution is a single interval of width 2c/|a|; here that width is 9, but the two strict endpoints (−1 and 8 themselves are excluded) trim it to 8 integers.
Add 7 throughout: −2 < 2x < 16.
Divide by 2: −1 < x < 8.
The integers strictly between −1 and 8 are 0, 1, 2, 3, 4, 5, 6, 7 — that is 8 integers.
Tip: For |ax − b| < c (c > 0), the solution is a single interval of width 2c/|a|; here that width is 9, but the two strict endpoints (−1 and 8 themselves are excluded) trim it to 8 integers.
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