UGC NET Questions (Paper – 1)

Solving 2x + 3 ≤ 11, we subtract 3 from both sides to get 2x ≤ 8. Dividing both sides by 2 yields x ≤ 4. On the real number line, this corresponds to all numbers less than or equal to 4. In interval notation, that set is written as (−∞, 4], with a square bracket at 4 indicating that 4 is included.

Option A:
Option A, (−∞, 4), excludes 4 itself and would represent x < 4, which is a strict inequality, not the given ≤ relation. Option B: Option B correctly captures x ≤ 4 by including all values up to and including 4, thus satisfying the inequality exactly. Option C: Option C, (−∞, 7], includes many values greater than 4, violating the upper bound imposed by 2x ≤ 8. Option D: Option D, [4, ∞), represents x ≥ 4, the opposite direction of the inequality, and only includes 4 and greater values.

Statements A and B are correct because adding the same number or multiplying by a positive number does not change the inequality direction. Statement C is also correct as multiplying by a negative number reverses the inequality sign, a key exam point. Statement E is true since solutions of such inequalities are commonly expressed as intervals or rays on the real line. D is false because if a > b and b > c, then a > c, not a < c. Therefore A, B, C and E only form the correct set. Option A: Option A is incomplete since it leaves out C and E. Without mentioning the effect of multiplying by a negative and the interval representation, it does not give a full picture of inequality properties. Option B: Option B is also incomplete because it omits E and thereby ignores the standard graphical representation of solution sets on the number line. Option C: Option C is correct as it combines the basic algebraic rules for manipulating inequalities with the interval depiction of solutions, while excluding D, which contradicts the transitivity of the “greater than” relation. Option D: Option D is wrong since it includes D, which reverses the transitive relation, and thus mixes a clearly false statement with true ones.