Q6. - Algebra 1

Question

State whether each sentence is true or false. If false, replace the underlined word, phrase, expression, or number to make a true sentence.

 

The equation $\frac{3x}{x-2}=\frac{6}{x-2}$ has an extraneous solution, 2.

Step-by-Step Solution

Answer

The given statement is True.

Step 1. State the concept of extraneous solution.

When a solution of a rational equation results in a zero in the denominator, that solution must be excluded. Such solutions are called extraneous solutions.

Step 2. Explanation of the statement.

The given equation is $\frac{3x}{x-2}=\frac{6}{x-2}$. On solving further, the solution thus obtained is:

$$\begin{gathered} \frac{3x}{x-2}=\frac{6}{x-2} \\ 3x=6 \\ x=2 \end{gathered}$$

Step 3. Write whether the statement is true or false.

But $x=2$ results in a zero in the denominator of the original equation. Therefore, $x=2$ is an extraneous solution. 

The given statement “The equation $\frac{3x}{x-2}=\frac{6}{x-2}$ has an extraneous solution, 2.” is true.