NinetynineCAT 2025 Slot 2 QA Q14 Inequalities & Modulus
Question 14 of the Quantitative Ability (QA) section, CAT 2025 Slot 2, with CAT's official answer and a worked solution.
The set of all real values of \( x \) for which \( \left( x ^ { 2 } - | x + 9 | + x \right) > 0 \), is
- A\( ( - \infty , - 9 ) \cup ( 9 , \infty ) \)
- B\( ( - \infty , - 3 ) \cup ( 3 , \infty ) \)
- C\( ( - 9 , - 3 ) \cup ( 3 , \infty ) \)
- D\( ( - \infty , - 9 ) \cup ( 3 , \infty ) \)
Answer: (B) ( - ∞ , - 3 ) ( 3 , ∞ ) Inequalities & Modulus MCQ
Solution
- For \(x \ge -9\): \(x^2 + x > x + 9 \Rightarrow x^2 > 9 \Rightarrow x < -3\) or \(x > 3\). This gives \([-9, -3) \cup (3, \infty)\).
- For \(x < -9\): \(x^2 + x > -x - 9 \Rightarrow x^2 + 2x + 9 > 0\), always true (discriminant negative).
- Union: \((-\infty, -3) \cup (3, \infty)\).
Shortcut. Test \(x = -10\): \(100 - 1 - 10 = 89 > 0\), so the region below \(-9\) is included; this leaves B or D, and \(x = -5\): \(25 - 4 - 5 = 16 > 0\) picks B.