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Graph the solution to the following inequality on the number line.(x - 2)(x-7)=0

Graph the solution to the following inequality on the number line.(x - 2)(x-7)=0-example-1
Graph the solution to the following inequality on the number line.(x - 2)(x-7)=0-example-1
Graph the solution to the following inequality on the number line.(x - 2)(x-7)=0-example-2
User Sse
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We have the following expression:


(x-2)\cdot(x-7)\leq0

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.


\begin{gathered} (x-2)\leq0 \\ \text{and} \\ (x-7)\ge0 \end{gathered}

This is equivalent to have:


\begin{gathered} x\leq2 \\ \text{and} \\ x\ge7 \end{gathered}

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.


\begin{gathered} (x-2)\ge0 \\ \text{and} \\ (x-7)\leq0 \end{gathered}

This is equivalent to have:


\begin{gathered} x\ge2 \\ \text{and} \\ x\leq7 \end{gathered}

In this case, the inequalities are compatible and the set or range of values of x that satisfies these inequalities are:


2\leq x\leq7

Plotting this in the graph:

Graph the solution to the following inequality on the number line.(x - 2)(x-7)=0-example-1
User Singingwolfboy
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