Question

Consider the following statements
Statement I The solution set of $2 \leq 3 x-4 \leq 5$ is $(2,3)$
Statement II $(-80,-10)$ is the solution set of $-12<4-\frac{3 x}{-5} \leq 2$
Choose the correct option.

• A. Statement I is true
• B. Statement II is true
• C. Both are true
• D. Both are false

Answer 1

Answer: (D)

I. The given inequality $2 \leq 3 x-4 \leq 5$
$\Rightarrow 2+4 \leq 3 x \leq 5+4$
$\Rightarrow 6 \leq 3 x \leq 9$
Dividing by 3 in each term,
$\frac{6}{3} \leq \frac{3 x}{3} \leq \frac{9}{3} $
$\Rightarrow 2 \leq x \leq 3$
$\therefore$ Solution set is $[2,3]$.
II. The given inequality
$ -12<4-\frac{3 x}{-5} \leq 2$
$\Rightarrow -12<4+\frac{3 x}{5} \leq 2$
Adding $(-4)$ to each term,
$ -12-4< 4+\frac{3 x}{5}-4 \leq 2-4$
$\Rightarrow -16 <\frac{3 x}{5} \leq-2$
Multiplying by $\frac{5}{3}$ to each term,
$ -16 \times \frac{5}{3}<\frac{3 x}{5} \times \frac{5}{3} \leq-2 \times \frac{5}{3}$
$\Rightarrow-\frac{80}{3}< x \leq-\frac{10}{3}$
$\therefore \text { Solution set is } \left.\left.\left(-\frac{80}{3},-\frac{10}{3}\right] \text { or }\right]-\frac{80}{3}, \frac{-10}{3}\right]$