Question

Which inequality is equivalent to B2%7D%7Bx%7D%20" id="TexFormula1" title="3+ \frac{4}{x} \geq \frac{x+2}{x} " alt="3+ \frac{4}{x} \geq \frac{x+2}{x} " align="absmiddle" class="latex-formula"> ?
A $\frac{2x+2}{x} \geq 0$
B $\frac{2x+6}{x} \geq 0$
C$\frac{-x+5}{x} \geq 0$
D$\frac{-x+9}{x} \geq 0$

Please show work and explain. I legitimately want to understand how to solve a question like this. Thank you!

Answer 1

First, take all expressions to one side of the inequality:

                 $\displaystyle{ 3+\frac{4}{x}- \frac{x+2}{x} \geq 0$.

Multiply 3 by $\displaystyle{ \frac{x}{x}$ to write it as a fraction with denominator equal to the other expressions

             $\displaystyle{ \frac{3x}{x}+\frac{4}{x}- \frac{x+2}{x} \geq 0$.


Since all three expressions have equal denominator, we collect them into one rational expression as follows:

$\displaystyle{ \frac{3x+4-x-2}{x}$, which is equal to $\displaystyle{ \frac{2x+2}{x}$.


Thus the inequality is $\displaystyle{ \frac{2x+2}{x} \geq 0$.


Answer: A