Question

Provide a counterexample to show that each statement is false.

Statement: If $xy>5y$, than $x>5$.

Short answer

The counterexample, $x=3$and $y=-6$, shows that the statement is false.

Step-by-step solution

Step 1. Define counterexample.

The counterexample is an example for which a hypothesis is true but the conclusion is false for an if-then statement.

Step 2. State the counterexample

In order to show, “If $xy>5y$, then $x>5$.” is false let$x=3$ and$y=-6$.

Step 3. Verify the counterexample.

If $x=3$and $y=-6$, then

$\begin{array}{c}xy=\left(3\right)\left(-6\right)\\ =\left(-18\right)\end{array}$

and,

$\begin{array}{c}5y=5\times \left(-6\right)\\ =\left(-30\right)\end{array}$

Since, $\left(-18\right)\text{>}\left(-30\right)$, hence $xy>5y$. But, $x=3<5$, thus, $x<5$,which contradicts the inequality.

Therefore, the counterexample, $x=3$and $y=-6$, shows that the statement is false.