Question

For the following​ equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in​ mind, solve the equation.StartFraction 2 Over 3 x EndFraction plus one fourth equals StartFraction 67 Over 6 x EndFraction minus one third

Answer 1

To solve the equation and find the restrictions on the variable, we begin with the equation given:

\[
\frac{2}{3x} + \frac{1}{4} = \frac{67}{6x} - \frac{1}{3}
\]

### Part a: Find Restrictions
The restrictions occur when any denominator in the equation is equal to zero. We have two denominators: \(3x\) and \(6x\).

- For \(3x\),
\[
3x = 0 \implies x = 0
\]

- For \(6x\),
\[
6x = 0 \implies x = 0
\]

Thus, the unique restriction on \(x\) is:
\[
x \neq 0
\]

### Part b: Solve the Equation
Now, we can solve the equation, keeping the restriction in mind.

1. Combine the fractions. To eliminate the denominators, we will find a common denominator, which is \(12x\). Rewrite each term with the common denominator:
\[
\frac{2}{3x} = \frac{8}{12x}, \quad
\frac{1}{4} = \frac{3x}{12x}, \quad
\frac{67}{6x} = \frac{134}{12x}, \quad
\frac{1}{3} = \frac{4x}{12x}
\]

2. Substitute these back into the equation:
\[
\frac{8}{12x} + \frac{3x}{12x} = \frac{134}{12x} - \frac{4x}{12x}
\]

3. Combine the fractions:
\[
\frac{8 + 3x}{12x} = \frac{134 - 4x}{12x}
\]

4. Since the denominators are the same and \(x \neq 0\), we can set the numerators equal to each other:
\[
8 + 3x = 134 - 4x
\]

5. Solve for \(x\):
- First, add \(4x\) to both sides:
\[
8 + 3x + 4x = 134
\]
\[
8 + 7x = 134
\]

- Now, subtract 8 from both sides:
\[
7x = 134 - 8
\]
\[
7x = 126
\]

- Finally, divide by 7:
\[
x = \frac{126}{7} = 18
\]

### Conclusion:
The value of \(x\) that solves the equation is:
\[
x = 18
\]

**Restrictions:** \(x \neq 0\) (the solution does not violate this restriction).