Which of the following is the solution of the inequality (x² + |4x + 3|) / (4 − 3x) ≥ 0 ?
A) each real x with x > 4/3
B) each real x with x < 4/3
C) each real x with x ≥ 4/3
D) each real x with x ≤ 4/3
E) each real x with x ≠ −3/4 ∧ x ≠ 0
IMAT 2024 · Question 51
Q51 - Algebraic inequalities
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The correct answer is B) each real x with x < 4/3.
The numerator, x² + |4x + 3|, can never be negative, and it can't be zero either, since that would need x = 0 and x = −3/4 at the same time. So it's always positive.
That means only the denominator matters: 4 − 3x > 0 → x < 4/3. The equals sign isn't allowed, because x = 4/3 makes the denominator zero.
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