Question

What values of $x$ make the two expressions below equal?

$\frac{\left(x+3\right)\left(x+8\right)}{5\left(x+8\right)}=\frac{x+3}{5}$

• A. All real numbers.
• B. All real numbers except $-3$.
• C. All real numbers except $-8$.
• D. All real numbers except $-8$ and $-3$.

Answer 1

The correct option is A

All real numbers.

Explanation for the correct answer:

Cross multiplying the terms.

$\frac{\left(x+3\right)\left(x+8\right)}{5\left(x+8\right)}=\frac{x+3}{5}$

$$\begin{gathered} \Rightarrow 5\left(x+3\right)\left(x+8\right)=5\left(x+3\right)\left(5\right)\left(x+8\right) \\ \Rightarrow 5\left(x^2+8x+3x+24\right)=5\left(x^2+8x+3x+24\right) \end{gathered}$$

Since the terms on the left-hand side and the right-hand side are the same, hence we can conclude that the equation independent of $x$, and hence it holds true for all real numbers value of $x$.

Therefore, the correct answer is option $\left(A\right)$