Q37. - Algebra 1

Question

Example 4 Find a counterexample for each conditional statement.

If $x^2=1$ then x must equal 1.

Step-by-Step Solution

Answer

The statement that If $x^2=1,$ then $x$ must equal 1 is not true.

Step 1. Given Statement.

If $x^2=1,$ then $x$ must equal 1.

Step 2. Description of step.

To find a counterexample for the given statement, solve the given expression and find the value of $x$ as

$\begin{array}{l}{x}^2=1\\ \Rightarrow x=\pm \sqrt{1}\\ \Rightarrow x=\pm 1\\ \Rightarrow x=1\text{ and }x=-1\end{array}$

Step 3. Conclusion.

So $x$ has two values, 1 and -1, but according to the given statement, $x$ must be equal to 1.

Therefore, the statement If $x^2=1,$ then $x$ must equal 1 is not true, and the counterexample is $x=-1$.