Question

If a variable $x$ satisfy the inequality $-2x+3\ge 11$ and $-x-5\le 7,$ then

• A. $4\le x\le 12$
• B. $-4\le x\le 12$
• C. $-12\le x\le -4$
• D. $-12\le x\le 4$

Answer 1

The correct option is C $-12\le x\le -4$

Given, $-2x+3\ge 11$

Subtract both sides of the inequality by 3
$-2x+3-3\ge 11-3$
$\Rightarrow -2x\ge 8$

Divide both sides of the inequality by -2 and reverse the inequality.
$\frac{-2x}{-2}\le \frac{8}{-2}$
$\Rightarrow x\le -4\dots \left(i\right)$
$\Rightarrow x\text{is less than or equal to}-4$

Also, $-x-5\le 7$

Add both sides of the inequality by 5
$-x-5+5\le 7+5$
$\Rightarrow -x\le 12$

Multiply both sides of the inequality by -1 and reverse the inequality.
$-x\times -1\ge 12\times -1$
$\Rightarrow x\ge -12\dots \left(ii\right)$
$\Rightarrow x\text{is greater than or equal to}-12$

Combining $\left(i\right)$ and $\left(ii\right),$ we get
$-12\le x\le -4$
$\Rightarrow x\text{is greater than or equal to}-12\text{and less than or equal to}-4$