Answer
False statement
Work Step by Step
$\left( x+5 \right)\left( x-4 \right)\ge 0$
Now to find the intervals, set the inequality to 0.
$\left( x+5 \right)\left( x-4 \right)=0$
The solution is,
$\begin{align}
& x+5=0 \\
& x=-5
\end{align}$
Or,
$\begin{align}
& x-4=0 \\
& x=4
\end{align}$
The roots divide the number line into three intervals: A, B and C.
Now, check whether the inequality $\left( x+5 \right)\left( x-4 \right)\ge 0$ is satisfied or not.
Take a test value from each interval and plug that into the equation as,
For interval A,
Test for $x=-6$
$\begin{align}
& \left( -6+5 \right)\left( -6-4 \right)=\left( -1 \right)\left( -10 \right) \\
& =10\ge 0
\end{align}$
So, in the interval A, the inequality is satisfied.
For interval B,
Test for $x=0$
$\begin{align}
& \left( 0+5 \right)\left( 0-4 \right)=\left( 5 \right)\left( -4 \right) \\
& =-20\not{\ge }0
\end{align}$
So, in the interval B, the inequality is not satisfied.
For interval C,
Test for $x=5$
$\begin{align}
& \left( 5+5 \right)\left( 5-4 \right)=\left( 10 \right)\left( 1 \right) \\
& =10\ge 0
\end{align}$
So, in the interval C, the inequality is satisfied.
So, from the above test, it is clear that the inequality $\left( x+5 \right)\left( x-4 \right)\ge 0$ is satisfied in the interval A: $\left( -\infty ,-5 \right)$ and B: $\left( 4,\infty \right)$ . The inequality symbol $\le $ means that the end points of the intervals is also included in the solution.
Therefore, the solution of the inequality $\left( x+5 \right)\left( x-4 \right)\ge 0$ is $\left( -\infty ,-5 \right]\bigcup \left[ 4,\infty \right)$.
