Answer
True statement
Work Step by Step
$\left( x-3 \right)\left( x+2 \right)\le 0$
Now to find the intervals, set the inequality to 0.
$\left( x-3 \right)\left( x+2 \right)=0$
The solution is,
$\begin{align}
& x-3=0 \\
& x=3
\end{align}$
Or,
$\begin{align}
& x+2=0 \\
& x=-2
\end{align}$
The roots divide the number line into three intervals: A, B and C.
Now, check whether the inequality $\left( x-3 \right)\left( x+2 \right)\le 0$ is satisfied or not.
Take a test value from each interval and plug that into the equation as,
For interval A,
Test for $x=-5$
$\begin{align}
& \left( -5-3 \right)\left( -5+2 \right)=\left( -8 \right)\left( -3 \right) \\
& =24\not{\le }0
\end{align}$
So, in the interval A, the inequality is not satisfied.
For interval B,
Test for $x=0$
$\begin{align}
& \left( 0-3 \right)\left( 0+2 \right)=\left( -3 \right)\left( 2 \right) \\
& =-6\le 0
\end{align}$
So, in the interval B, the inequality is satisfied.
For interval C,
Test for $x=5$
$\begin{align}
& \left( 5-3 \right)\left( 5+2 \right)=\left( 2 \right)\left( 7 \right) \\
& =14\not{\le }0
\end{align}$
So, in the interval C, the inequality is not satisfied.
So, from the above test, it is clear that the inequality $\left( x-3 \right)\left( x+2 \right)\le 0$ is satisfied only in the interval B: $\left( -2,3 \right)$. The inequality symbol $\le $ means that the end points of the intervals are also included in the solution.
Therefore, the solution of the inequality $\left( x-3 \right)\left( x+2 \right)\le 0$ is $\left[ -2,3 \right]$.
