Solution set of an inequality

[math_girl111]

I am having problems with this question:

For the inequality (x-4) / (x + 2) > 0 find its solution set.

I can graph the inequality and see that the graph points are above 0 when x is from -infinity to -2, but how do I explain it? How do you go about solving inequalities in general without looking at the graph because I am required to show complete work and show my reasoning when I give the answer.

Thanks for the help!

 

[soroban]

Hello, math_girl111!

\(\displaystyle \text{For the inequality: }\:\frac{x-4}{x + 2} \:>\: 0,\;\text{ find its solution set.}\)

\(\displaystyle \text{A fraction is positive if: }\:\begin{Bmatrix}\text{[1] both the numerator and denominator are positive.} \\ \text{or} \\ \text{[2] both the numerator and denominator are negative.} \end{Bmatrix}\)


\(\displaystyle \text{Case 1: }\;\begin{Bmatrix} x-4 \:>\:0 & \Rightarrow & x \:>\:4 \\ x+2 \:>\:0 & \Rightarrow & x \:>\:\text{-}2 \end{Bmatrix}\)

. . \(\displaystyle \text{Both inequalties are satisfied when }x \:>\:4.\)


\(\displaystyle \text{Case 2: }\:\begin{Bmatrix}x-4 \:<\:0 & \Rightarrow & x \:<\:4 \\ x+2 \:<\<0 & \Rightarrow & x \:<\:\text{-}2 \end{Bmatrix}\)

. . \(\displaystyle \text{Both inequalities are satisfied when }x \:<\:\text{-}2.\)


\(\displaystyle \text{The solution set is: }\:\begin{Bmatrix} (\text{-}\infty,\,\text{-}2) \:\cup (4,\,\infty) \\ \\[-3mm] x \:<\:\text{-}2\:\text{ or }\:x \:>\:4 \end{Bmatrix}\)