Numerical answer
How many integers x satisfy both |x − 4| > 1 and |x − 4| < 7?
Show answer and solution
Answer: 10
|x − 4| < 7 means −3 < x < 11, i.e. x ∈ {−2, −1, 0, …, 10} (13 integers).
|x − 4| > 1 means x < 3 or x > 5, which excludes x = 3, 4, 5 from that range.
Remaining integers: {−2, −1, 0, 1, 2} ∪ {6, 7, 8, 9, 10} — that's 5 + 5 = 10 integers.
Tip: Solve each modulus inequality separately into a simple range/exclusion first, then intersect the integer sets — don't try to combine the two moduli algebraically.
|x − 4| > 1 means x < 3 or x > 5, which excludes x = 3, 4, 5 from that range.
Remaining integers: {−2, −1, 0, 1, 2} ∪ {6, 7, 8, 9, 10} — that's 5 + 5 = 10 integers.
Tip: Solve each modulus inequality separately into a simple range/exclusion first, then intersect the integer sets — don't try to combine the two moduli algebraically.
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