Question

Which inequality represents all values of × for which the quotient below is lefined?

Answer 1

SOLUTION

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the given expression

$\sqrt{28(x-1)}\div\sqrt{8x^2}$

STEP 2: Simplify the expression

For the quotient to be defined, the numerator and the denominator must be greater than zero, this means that:

$\begin{gathered} \sqrt{8x^2}>0 \\ \mathrm{Square\:both\:sides} \\ \left(\sqrt{8x^2}\right)^2>0^2 \\ 8x^2>0 \\ x<0\quad \mathrm{or}\quad \:x>0 \end{gathered}$

For the numerator,

$\begin{gathered} \mathrm{Square\:both\:sides} \\ \left(\sqrt{28\left(x-1\right)}\right)^2>0^2 \\ \mathrm{Simplify} \\ 28x-28>0 \\ x>1 \\ \mathrm{Combine\:the\:intervals} \\ x>1\quad \mathrm{and}\quad \:x\ge \:1 \\ x>1 \end{gathered}$

Merging both interval, we have:

$\begin{gathered} x<0,x>0,x>1 \\ x>1 \end{gathered}$

Hence, the answer is given as:

$x>1$