For how many integer values of x is |x − 3| + |x + 1| ≤ 6?
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Answer & Explanation
Correct answer
C
Method 1 (region analysis): the sum |x − 3| + |x + 1| equals the total distance from x to 3 and to −1. For −1 ≤ x ≤ 3 the sum is the fixed gap 4, which is ≤ 6. For x > 3 the sum is 2x − 2, and 2x − 2 ≤ 6 gives x ≤ 4. For x < −1 the sum is 2 − 2x, and 2 − 2x ≤ 6 gives x ≥ −2. Combining, −2 ≤ x ≤ 4, whose integers are −2, −1, 0, 1, 2, 3, 4, a total of 7.
Method 2 (test the boundaries): at x = −2 the sum is 5 + 1 = 6 and at x = 4 the sum is 1 + 5 = 6, both meeting ≤ 6, while x = −3 and x = 5 give 7 > 6; so the included integers run −2 through 4, which is 7 values.
Choice 5 ignores the tails. Choice 6 undercounts an endpoint. Choice 8 mis-places the lower boundary. Choice 9 extends both tails too far.