Answer
True statement
Work Step by Step
$\left( x-1 \right)\left( x-6 \right)>0$
Now to find the intervals, set the inequality to 0.
$\left( x-1 \right)\left( x-6 \right)=0$
The solution is,
$\begin{align}
& x-1=0 \\
& x=1
\end{align}$
Or,
$\begin{align}
& x-6=0 \\
& x=6
\end{align}$
The roots divide the number line into three intervals: A, B and C.
Now, check whether the inequality $\left( x-1 \right)\left( x-6 \right)>0$ is satisfied or not.
Take a test value from each interval and plug that into the equation as,
For interval A,
Test for $x=0$
$\begin{align}
& \left( 0-1 \right)\left( 0-6 \right)=\left( -1 \right)\left( -6 \right) \\
& =6>0
\end{align}$
So, in the interval A, the inequality is satisfied.
For interval B,
Test for $x=2$
$\begin{align}
& \left( 2-1 \right)\left( 2-6 \right)=\left( 1 \right)\left( -4 \right) \\
& =-4\not{>}0
\end{align}$
So, in the interval B, the inequality is not satisfied.
For interval C,
Test for $x=7$
$\begin{align}
& \left( 7-1 \right)\left( 7-6 \right)=\left( 6 \right)\left( 1 \right) \\
& =6>0
\end{align}$
So, in the interval C, the inequality is not satisfied.
So, from the above test, it is clear that the inequality $\left( x-1 \right)\left( x-6 \right)>0$ is satisfied in the interval the interval A: $\left( -\infty ,1 \right)$ and B: $\left( 6,\infty \right)$. The inequality symbol $>$ implies that $x=1\text{ and 6}$ are not included in the solution.
Therefore, the solution of the inequality $\left( x-1 \right)\left( x-6 \right)>0$ is $\left( -\infty ,1 \right)\bigcup \left( 6,\infty \right)$ or $\left\{ \left. x \right|x<1\text{ or }x>6 \right\}$.
