UGC NET Questions (Paper – 1)

A, B and C are correct standard facts: absolute value measures distance, ∣x∣ < 3 implies −3 < x 4 implies x 4. D is false because absolute value cannot be negative, so ∣x∣ = −2 has no real solution. E is false because ∣x∣ = 5 has two real solutions, x = 5 and x = −5, not exactly one. Therefore, the correct set is A, B and C only.

Option A:
Option A is incomplete because it omits C, which is also a correct rule for solving strict “greater than” absolute value inequalities.

Option B:
Option B is incorrect because it includes E, which is false since ∣x∣ = 5 has two solutions, not one.

Option C:
Option C is correct because it lists exactly the true statements (A, B, C) and excludes both false statements (D, E).

Option D:
Option D is incorrect because it includes D (which is impossible) and E (which is also false), so it mixes false statements with true ones.

To solve the inequality 2x − 5 > 7, we first add 5 to both sides, giving 2x > 12. Then we divide both sides by 2, which is positive, so the inequality direction remains the same, yielding x > 6. Thus, the solution set consists of all real numbers greater than 6, and this is the equivalent inequality in terms of x.

Option A:
Option A captures the full result of correctly isolating x. It incorporates both the addition and division steps and expresses the final constraint directly as x > 6.

Option B:
Option B, x > 5, would follow only if the inequality were 2x − 5 > 5, not 2x − 5 > 7. It represents a weaker condition than required.

Option C:
Option C, x > 4, ignores more of the numerical shift and significantly broadens the solution set beyond what the original inequality allows.

Option D:
Option D, x > 3, is even looser and does not align with the operations needed to isolate x in the given inequality.