# How do you solve (4x)/(x+7)<x?

Jul 1, 2016

$x > 0$ or $x \in \left(- 3 , - 7\right)$

#### Explanation:

Note: for the given expression to be meaningful $x \ne - 7$
and $x \ne 0$ (since this would make both sides equal).

Case 1: $\textcolor{b l a c k}{x > 0}$

$\textcolor{w h i t e}{\text{XXX}} \frac{4 x}{x + 7} < x$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow \frac{4}{x + 7} < 1$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow 4 < x + 7$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow x > - 3$

$\textcolor{w h i t e}{\text{XXX}}$but by the Case 1 limitation $x > 0$ (which is more limiting).

Case 2: $\textcolor{b l a c k}{x < 0}$

$\textcolor{w h i t e}{\text{XXX}} \frac{4 x}{x + 7} < x$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow \frac{4}{x + 7} > 1$ (dividing by a negative reverses the inequality)

color(white)("XXX"){: (color(black)("Case 2a: "),color(white)("XX"),color(black)("Case 2b: ")), (color(white)("X")color(black)(x < -7),,color(white)("X")color(black)(x > -7)), (color(white)("XX")4/(x+7) > 1,,color(white)("XX")4/(x+7) > 1), (color(white)("XX")rarr 4 < x+7,,color(white)("XX")4 > x+7), (color(white)("XX")rarr -3 < x,,color(white)("XX")-3 > x), (color(white)("XX")"impossible; since ",,color(white)("XX")rarr x in (-7,-3)), (color(white)("XX")color(white)("XXXX")x < 7,,) :}

The following graph image might help understand this relationship.
(Note the asymptote for $\textcolor{red}{\frac{4 x}{x + 7}}$)

The region for which $\textcolor{red}{\frac{4 x}{x + 7}} < \textcolor{b l u e}{x}$ is shaded in $\textcolor{g r e e n}{\text{green}}$