Multiple choice

Solve the following inequalities: 3| x – 1| + x2 – 7 > 0

  1. x > -1

  2. x < - 1

  3. x > 2

  4. Both (b) and (c)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Solve 3|x-1| + x^2 - 7 > 0. Case 1: x >= 1, 3(x-1) + x^2 - 7 > 0 -> x^2 + 3x - 10 > 0 -> (x+5)(x-2) > 0. For x >= 1, this holds for x > 2. Case 2: x < 1, 3(1-x) + x^2 - 7 > 0 -> x^2 - 3x - 4 > 0 -> (x-4)(x+1) > 0. For x < 1, this holds for x < -1. Combining gives x < -1 or x > 2.

AI explanation

The given inequality is 3 times the absolute value of (x minus 1) plus x squared minus 7 is greater than 0. Since the absolute value is always non-negative, 3 times the absolute value of (x minus 1) is always greater than or equal to 0. Furthermore, for any real number x, x squared is always greater than or equal to 0. By analyzing the regions, when x is less than -1, the expression x squared minus 7 is positive and dominates, satisfying the inequality; when x is greater than 2, the same condition holds true. Therefore, both x less than -1 and x greater than 2 satisfy the inequality.

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