UGCNET Questions

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.

The condition x > 3 tells us that x is strictly greater than 3 on the number line. Adding the same number to both sides of an inequality preserves its direction, so adding 2 to get x + 2 > 3 + 2 shows that x + 2 > 5. This relationship must hold for every x satisfying the original inequality.

Option A:
Option A follows directly from adding 2 to both sides of the original inequality. Since 3 + 2 equals 5, every value of x greater than 3 will make x + 2 greater than 5 as well.

Option B:
Option B, x − 5 > 0, would require x to be greater than 5, which is stronger than the given condition. Since x may be between 3 and 5, this statement need not always be true.

Option C:
Option C, 2x < 6, implies x < 3, which actually contradicts the assumption that x is greater than 3. Option D: Option D, x/2 < 1.5, is another form of x < 3 and thus conflicts with the original inequality rather than following from it.

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.