Match List-I with List-II
| List-I | List-II | ||
| Inequalities | Solution set | ||
| (A) | If \( |3-5x|\lt 7 \), then \( x\in \) | (I) | \( \left[-\displaystyle\frac{5}{2},\infty\right) \) |
| (B) | If \( 3x-7\gt 5x-1 \), then \( x\in \) | (II) | \( (-\infty,-3) \) |
| (C) | If \( \displaystyle\frac{x}{3}\gt\displaystyle\frac{x}{2}+1 \), then \( x\in \) | (III) | \( \left(-\displaystyle\frac{4}{5},2\right) \) |
| (D) | If \( 5x+4\le 7x+9 \), then \( x\in \) | (IV) | \( (-\infty,-6) \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (III), (B) - (II), (C) - (IV), (D) - (I)
(A) \( |3-5x|\lt 7 \)
This matches (III) \( \left(-\frac{4}{5},2\right) \).
\( \displaystyle -7\lt 3-5x\lt 7 \)
\( \displaystyle -10\lt -5x\lt 4 \)
Dividing by −5 reverses the inequality signs:
\( \displaystyle 2\gt x\gt -\frac{4}{5} \)
\( \displaystyle x\in\left(-\frac{4}{5},2\right) \)
(B) \( 3x-7\gt 5x-1 \)
This matches (II) \( (-\infty,-3) \).
\( \displaystyle -6\gt 2x \)
\( \displaystyle x\lt -3 \)
(C) \( \frac{x}{3}\gt\frac{x}{2}+1 \)
This matches (IV) \( (-\infty,-6) \).
Multiplying by 6 (positive, so the sign stays):
\( \displaystyle 2x\gt 3x+6 \)
\( \displaystyle -x\gt 6 \)
\( \displaystyle x\lt -6 \)
(D) \( 5x+4\le 7x+9 \)
This matches (I) \( \left[-\frac{5}{2},\infty\right) \).
\( \displaystyle -5\le 2x \)
\( \displaystyle x\ge -\frac{5}{2} \)
The endpoint is included because the inequality is ≤, which gives the square bracket.
So the matching is (A)-(III), (B)-(II), (C)-(IV), (D)-(I).