Solve the Fraction Equation: Find X in -x + 1/4 - 1/3 + 1/8 = 5 + 1/4x - 1/3x

To solve this equation, we will follow these steps:

• Step 1: Simplify each side of the equation by combining like terms.
• Step 2: Find and apply the least common denominator to eliminate fractions.
• Step 3: Solve the resulting linear equation.

Step 1: Simplify both sides of the equation. The original equation is:

$-x + \frac{1}{4} - \frac{1}{3} + \frac{1}{8} = 5 + \frac{1}{4}x - \frac{1}{3}x$

On the left side, combine the constant terms:

$\frac{1}{4} - \frac{1}{3} + \frac{1}{8}$

The least common denominator (LCD) for these fractions is 24.

$\frac{1}{4} = \frac{6}{24}, \quad \frac{1}{3} = \frac{8}{24}, \quad \frac{1}{8} = \frac{3}{24}$

Combine them:

$\frac{6}{24} - \frac{8}{24} + \frac{3}{24} = \frac{1}{24}$

The left side of the equation becomes:

$-x + \frac{1}{24}$

On the right side, combine the $x$ terms:

$\frac{1}{4}x - \frac{1}{3}x$

Express with the common denominator 12:

$\frac{1}{4}x = \frac{3}{12}x, \quad \frac{1}{3}x = \frac{4}{12}x$

Combine them:

$\frac{3}{12}x - \frac{4}{12}x = -\frac{1}{12}x$

The right side of the equation becomes:

$5 - \frac{1}{12}x$

Step 2: Combine the aligned equation:

$-x + \frac{1}{24} = 5 - \frac{1}{12}x$

Step 3: Eliminate fractions by multiplying the entire equation by 24 (the LCD of the denominators 1, 24, 12):

$24(-x + \frac{1}{24}) = 24(5 - \frac{1}{12}x)$

Simplifies to:

$-24x + 1 = 120 - 2x$

Rearrange to solve for $x$:

• Get all $x$ terms on one side:
$-24x + 2x = 120 - 1$ $-22x = 119$
• Divide both sides by -22:
$x = -\frac{119}{22}$

In conclusion, the value of $x$ is:

$-\frac{119}{22}$