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 25 Over 6 x EndFraction minus one third

Answer 1

To solve the equation \(\frac{2}{3x} + \frac{1}{4} = \frac{25}{6x} - \frac{1}{3}\), we will perform the following steps:

### Step a: Identify restrictions on the variable

The equations have denominators that can become zero if the variable \(x\) takes certain values. We look for any term in the denominator:

1. For \(\frac{2}{3x}\), the denominator is \(3x\). This term is undefined when \(3x = 0\) which gives \(x = 0\).
2. For \(\frac{25}{6x}\), the denominator is \(6x\). This term is also undefined when \(6x = 0\), which also gives \(x = 0\).

Thus, the restriction on the variable is:
\[
x \neq 0
\]

### Step b: Solve the equation

We now proceed to solve the equation while keeping the restriction in mind.

1. Start with the original equation:
\[
\frac{2}{3x} + \frac{1}{4} = \frac{25}{6x} - \frac{1}{3}
\]

2. To eliminate the fractions, find a common denominator for all terms. The least common multiple (LCM) of the denominators \(3x\), \(4\), and \(6x\) is \(12x\). So we will multiply each term by \(12x\):
\[
12x \cdot \frac{2}{3x} + 12x \cdot \frac{1}{4} = 12x \cdot \frac{25}{6x} - 12x \cdot \frac{1}{3}
\]

3. Simplifying each term:
- For the first term: \(12x \cdot \frac{2}{3x} = 4 \cdot 2 = 8\)
- For the second term: \(12x \cdot \frac{1}{4} = 3x\)
- For the third term: \(12x \cdot \frac{25}{6x} = 2 \cdot 25 = 50\)
- For the fourth term: \(12x \cdot \frac{1}{3} = 4x\)

Putting it all together:
\[
8 + 3x = 50 - 4x
\]

4. Combine like terms. Start by adding \(4x\) to both sides:
\[
8 + 3x + 4x = 50
\]
This simplifies to:
\[
8 + 7x = 50
\]

5. Subtract \(8\) from both sides:
\[
7x = 50 - 8
\]
Simplifying gives:
\[
7x = 42
\]

6. Finally, divide both sides by \(7\):
\[
x = \frac{42}{7} = 6
\]

### Final Answer

**a.** The restriction on the variable is \(x \neq 0\).

**b.** The solution to the equation is \(x = 6\).