Solve the Fraction Equation: -1/5x + 1/3 - 1/4x + 1 = 3x - 1/5

To solve this problem, we'll proceed with the following steps:

• Step 1: Combine like terms on both sides of the equation.
• Step 2: Eliminate the fractions by finding a common multiple.
• Step 3: Solve for $x$ by isolating it on one side.

Now, let's work through each step:
**Step 1**: Combine like terms.
On the left side: Combine $-\frac{1}{5}x$ and $-\frac{1}{4}x$:
$-\frac{1}{5}x - \frac{1}{4}x = -\left(\frac{4}{20}x + \frac{5}{20}x\right) = -\frac{9}{20}x$.
The equation becomes: $-\frac{9}{20}x + \frac{1}{3} + 1 = 3x - \frac{1}{5}$.

**Step 2**: Eliminate fractions by multiplying the whole equation by the least common multiple (LCM) of the denominators (20, 3, 5).
The LCM of 20, 3, and 5 is 60.
Multiplying each term by 60 gives:
$60\left(-\frac{9}{20}x\right) + 60\left(\frac{1}{3}\right) + 60 \times 1 = 60 \times 3x - 60\left(\frac{1}{5}\right)$
This simplifies to:
$-27x + 20 + 60 = 180x - 12$.

**Step 3**: Combine constants and isolate $x$.
Combine constants on the left side: $-27x + 80 = 180x - 12$.
Add $27x$ to both sides: $80 = 207x - 12$.
Add 12 to both sides: $92 = 207x$.
Divide both sides by 207: $x = \frac{92}{207}$.
Simplify $\frac{92}{207}$ to $\frac{4}{9}$ (as both 92 and 207 are divisible by 23).

Therefore, the solution to the problem is $x = \frac{4}{9}$.