Question

Match the following inequalities with their solutions.

• A. $x\le 2$
• B. $x\le -1$
• C. $x<-2$
• D. $x>5$

Answer 1

Concept: Multiplying or dividing both sides of an inequality with a negative number changes the sign of the inequality.

A. $3x+4\le 10$
$\Rightarrow 3x\le 10-4$
$\Rightarrow 3x\le 6$
Dividing by $3$ on both sides, we get,

$\Rightarrow \frac{3x}{3}\le \frac{6}{3}$

$\Rightarrow x\le 2$

B. $-2x+6\ge 8$

$\Rightarrow -2x\ge 8-6$

$\Rightarrow -2x\ge 2$

Dividing both sides by $2$, we get,

$\frac{-2x}{2}\ge \frac{2}{2}$

$-x\ge 1$
Now, multiplying with $-1$ on both sides, we get,

$\Rightarrow -x\times -1\le 1\times -1$

$\Rightarrow x\le -1$

C. $7<5-x$
$\Rightarrow 7-5<-x$

$\Rightarrow 2<-x$

Multiplying with $-1$ on both sides, we get,

$\Rightarrow 2\times -1>-x\times -1$

$\Rightarrow -2>x$

$\Rightarrow x<-2$

D. $-8>7-3x$

$\Rightarrow -8-7>-3x$

$\Rightarrow -15>-3x$

Dividing with $-3$ on both sides, we get

$\Rightarrow \frac{-15}{-3}<\frac{-3x}{-3}$

$\Rightarrow 5<x\Rightarrow x>5$