Answer
The solution of rational inequality $\frac{-x-2}{x-4}\ge 0$ is $\left\{ \left[ -2,4 \right) \right\}$ or $\left\{ x|-2\le x<4 \right\}$.
Work Step by Step
Firstly, write the inequality in the form $f\left( x \right)\ge 0$
The provided expression is already in the form $f\left( x \right)\ge 0$
Here, $f\left( x \right)=\frac{-x-2}{x-4}$
Now, put the numerator and denominator of the rational inequality $f\left( x \right)=\frac{-x-2}{x-4}$ equal to zero.
That is, $\begin{align}
& -x-2=0 \\
& x=-2 \\
\end{align}$
and $\begin{align}
& x-4=0 \\
& x=4 \\
\end{align}$
Now, solve the above expressions for x.
Thus, $x=-2,4$
Now, locate the point $x=-2,4$ on the number line.
There are three intervals $\left( -\infty ,-2 \right],\left[ -2,4 \right]\text{ and }\left[ 4,\infty \right)$.
Now, test any value in the interval and evaluate $f$ at that point.
Now, take value $-3$ from the interval $\left( -\infty ,-2 \right]$
Put $-3$ in place of x in the equation of $f\left( x \right)=\frac{-x-2}{x-4}$ ,
$\begin{align}
& f\left( -3 \right)=\frac{-\left( -3 \right)-2}{-3-4} \\
& =\frac{3-2}{-3-4} \\
& =\frac{1}{-7} \\
& =-\frac{1}{7}
\end{align}$
Thus, $f\left( x \right)<0$
Now, take value $1$ from the interval $\left[ -2,4 \right]$
Put $1$ in place of x in the equation of $f\left( x \right)=\frac{-x-2}{x-4}$ ,
$\begin{align}
& f\left( -1 \right)=\frac{-\left( -1 \right)-2}{-1-4} \\
& =\frac{1-2}{-1-4} \\
& =\frac{-1}{-5} \\
& =\frac{1}{5}
\end{align}$
Thus, $f\left( x \right)>0$
Now, take value $5$ from the interval $\left[ 4,\infty \right)$
Put $5$ in place of x in the equation of $f\left( x \right)=\frac{-x-2}{x-4}$ ,
$\begin{align}
& f\left( 5 \right)=\frac{-\left( 5 \right)-2}{5-4} \\
& =\frac{-7}{1} \\
& =-7
\end{align}$
Thus, $f\left( x \right)<0$
Since it is provided that the function is greater than or equal to zero, include the solution of $f\left( x \right)=0$ , which is obtained on simplifying the numerator.
So, the only interval is: $\left[ -2,4 \right)$ or $\left\{ x|-2\le x<4 \right\}$
