Problem 103 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}+3}{3}=x^{2}+1$$

Step-by-Step Solution

Answer

The statement \(\frac{x^{2}+3}{3}=x^{2}+1\) is False. The correct statement to make it True is \(\frac{x^{2}}{3}+1 = x^{2}+1\).

Step 1: Simplify the left side of the equation

Start by simplifying the left side of the equation, which involves a division. Divide each term in the numerator by 3. So, the simplified version of \(\frac{x^{2}+3}{3}\) becomes \(\frac{x^{2}}{3}+1\).

Step 2: Compare the left side with the right side

After simplifying the left side, it's time to compare it with the right side of the equation. If \(\frac{x^{2}}{3}+1 = x^{2}+1\), the original statement is true; otherwise, it's false.

Step 3: Correcting the false statement

After comparison, it's found that the original statement is false as \(\frac{x^{2}}{3}+1\) is not equal to \(x^{2}+1\). In order to make the statement true, the correct statement should be \(\frac{x^{2}}{3}+1 = x^{2}+1\).