We have the following expression:
[tex](x-2)\cdot(x-7)\leq0[/tex]
and we want to know the set of values of x that satisfies the inequality.
We see that in order to be the left-hand side of the inequality less or equal to zero we must have:
1) First case.
(x-2) is negative and (x-7) is positive, so its product is a negative number.
[tex]\begin{gathered} (x-2)\leq0 \\ \text{and} \\ (x-7)\ge0 \end{gathered}[/tex]
This is equivalent to have:
[tex]\begin{gathered} x\leq2 \\ \text{and} \\ x\ge7 \end{gathered}[/tex]
There are no values of x that can satisfy both inequalities at the same time. So this is an incompatible solution.
2) Second case
(x-2) is positive and (x-7) is negative, so its product is a negative number.
[tex]\begin{gathered} (x-2)\ge0 \\ \text{and} \\ (x-7)\leq0 \end{gathered}[/tex]
This is equivalent to have:
[tex]\begin{gathered} x\ge2 \\ \text{and} \\ x\leq7 \end{gathered}[/tex]
In this case, the inequalities are compatible and the set or range of values of x that satisfies these inequalities are:
[tex]2\leq x\leq7[/tex]
Plotting this in the graph:
