Question

Which of the following inequality implies that $x$ is less than or equal to $-2?$

• A. $-x\ge -2$
• B. $-x\le 2$
• C. $-x\le -2$
• D. $-x\ge 2$

Answer 1

The correct option is D $-x\ge 2$

$\left(A\right)-x\ge -2$
Multiply both the sides by $-1$ to make the coefficient of $x$ as 1.
Also, multiplying both sides of the inequality by a negative number changes the direction of the inequality.
$-x\times -1\le -2\times -1$
$\Rightarrow x\le 2$
$\Rightarrow x\text{is less than or equal to}2$

$\left(B\right)-x\le 2$
Multiply both the sides by $-1$ and reverse the inequality.
$-x\times -1\ge 2\times -1$
$\Rightarrow x\ge -2$
$\Rightarrow x\text{is greater than or equal to}-2$

$\left(C\right)-x\le -2$
Multiply both the sides by $-1$ and reverse the inequality.
$-x\times -1\ge -2\times -1$
$\Rightarrow x\ge 2$
$\Rightarrow x\text{is greater than or equal to}2$

$\left(D\right)-x\ge 2$
Multiply both the sides by $-1$ and reverse the inequality.
$-x\times -1\le 2\times -1$
$\Rightarrow x\le -2$
$\Rightarrow x\text{is less than or equal to}-2$

Hence, $-x\ge 2$ implies that $x$ is less than or equal to $-2.$