Question

Which inequality represents all values of x for which the quotient below is defined?A.-2 x 5B.x -2C.-2 x < 5D.x 5

Answer 1

The quotient is given to be:

$\sqrt{x+2}\div \sqrt{5-x}$

Rewrite the expression:

$\Rightarrow\frac{\sqrt{x+2}}{\sqrt{5-x}}=\sqrt{\frac{x+2}{5-x}}$

The quotient will be defined under the following condition:

$\frac{x+2}{5-x}\ge0$

The inequality will work if:

$x+2\ge0\text{ or }5-x\ge0$

Hence, we can get the interval to be:

$\begin{gathered} x\ge-2 \\ or \\ x\leq5 \end{gathered}$

The inequality will be undefined at the point:

$\begin{gathered} 5-x=0 \\ \therefore \\ x=5 \end{gathered}$

Therefore, the function will be defined over the interval:

$-2\leq x<5$

OPTION C is correct.