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).