Question

If $x$ satisfies the inequality $4-x^2\ge x-2$ , then $x$ lies in the interval:

• A. $x\ge -3$
• B. $-5\le x\le 0$
• C. $-3\le x\le 2$
• D. $x\le -3$and $x\ge -2$
• E. $-2\le x\le 3$

Answer 1

The correct option is C $-3\le x\le 2$

$4-x^2\ge x-2$

$4-x^2-x+2\ge 0$

$-x^2-x+6\ge 0$

$x^2+x-6\le 0$

$(x+3)(x-2)\le 0$

This means that either $(x+3)$ is negative or $(x-2)$ is negative.

When $(x+3)$ is negative and $(x-2)$ is positive

$x+3\le 0$ and $x-2\ge 0$

So, $x\le -3$ and $x\ge 2$

When $(x-2)$ is negative and $(x+3)$ is positive

$x-2\le 0$ and $x+3\ge 0$

So, $x\le 2$ and $x\ge -3$

So, $-3\le x\le 2$