Solve the Fraction Equation: Find X in 1/4 + 2/5x = -3/4 + 1/10x

To solve this equation, we will perform the following steps:

• Step 1: Move terms involving $x$ to one side of the equation. Subtract $\frac{1}{10}x$ from both sides:
$\frac{2}{5}x - \frac{1}{10}x = -\frac{3}{4} - \frac{1}{4}$
• Step 2: Simplify the equation:

First, simplify the right-hand side:

$-\frac{3}{4} - \frac{1}{4} = -\frac{4}{4} = -1$
• Step 3: Align terms with $x$. Find a common denominator for the fractions on the left side:

The common denominator for $\frac{2}{5}x$ and $\frac{1}{10}x$ is 10.

$\frac{4}{10}x - \frac{1}{10}x = \frac{3}{10}x$
• Step 4: Set up the simplified equation:
$\frac{3}{10}x = -1$
• Step 5: Solve for $x$ by multiplying both sides by the reciprocal of the fraction's coefficient:
$x = -1 \times \frac{10}{3} = -\frac{10}{3}$
• Step 6: Final answer check verifies against the multiple-choice option.

Therefore, the solution to the equation is $x = -\frac{10}{3}$.