Problem 105 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$6+\frac{1}{x}=\frac{7}{x}$$

Step-by-Step Solution

Answer

The statement \(6 + \frac{1}{x} = \frac{7}{x}\) is false. The correct equation is \(\frac{7 - \frac{1}{x}}{x} = \frac{7}{x}\)

Step 1: Simplify the Equation

First, remove common factors on both sides of the equation. \(6 + \frac{1}{x} = \frac{7}{x}\) can be rewritten as \(\frac{6x + 1}{x} = \frac{7}{x}\).

Step 2: Compare the Numerators

Since the denominators are the same, compare the numerators of the fractions on both sides of the equation. If they are equal, the original equation is true. If they are not equal, it is false. Here, we compare \(6x + 1\) and 7. They are not equal, which means the original equation is false.

Step 3: Alter the False Statement into a True Statement.

The original statement is false. To make it true, we can adjust the equation so that the numerators match. For instance, we can adjust 6 into 7-1/x, to make the left side equivalent to the right side. So, the correct equation is \(\frac{7 - \frac{1}{x}}{x} = \frac{7}{x}\)