Numerical answer
How many integers x satisfy both |x + 6| ≥ 4 and |x + 6| ≤ 9?
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Answer: 12
|x + 6| ≥ 4 means x + 6 ≥ 4 or x + 6 ≤ −4, i.e. x ≥ −2 or x ≤ −10.
|x + 6| ≤ 9 means −9 ≤ x + 6 ≤ 9, i.e. −15 ≤ x ≤ 3.
Intersecting: x ∈ [−15, −10] ∪ [−2, 3].
[−15, −10] has 6 integers; [−2, 3] has 6 integers. Total = 6 + 6 = 12.
Tip: Solve each side of a compound modulus inequality into a simple range or exclusion first, then intersect — never combine the two inequalities before simplifying.
|x + 6| ≤ 9 means −9 ≤ x + 6 ≤ 9, i.e. −15 ≤ x ≤ 3.
Intersecting: x ∈ [−15, −10] ∪ [−2, 3].
[−15, −10] has 6 integers; [−2, 3] has 6 integers. Total = 6 + 6 = 12.
Tip: Solve each side of a compound modulus inequality into a simple range or exclusion first, then intersect — never combine the two inequalities before simplifying.
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