Question

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

Answer 1

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

STEP 1: Write the given expression

`√(28(x-1))/√(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} √(8x^2)>0 \\ \mathrm{Square\:both\:sides} \\ \left(√(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(√(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`