The problem is to solve the following linear equation for $x$: $\frac{3}{10}x - \frac{3}{2} = \frac{4}{5}x + 1$

1. Problem Description

The problem is to solve the following linear equation for $x$:

$\frac{3}{10}x - \frac{3}{2} = \frac{4}{5}x + 1$

2. Solution Steps

First, we want to isolate terms with $x$ on one side of the equation and constant terms on the other side. Subtract $\frac{3}{10}x$ from both sides:

$-\frac{3}{2} = \frac{4}{5}x - \frac{3}{10}x + 1$

$-\frac{3}{2} = \frac{8}{10}x - \frac{3}{10}x + 1$

$-\frac{3}{2} = \frac{5}{10}x + 1$

$-\frac{3}{2} = \frac{1}{2}x + 1$

Next, subtract 1 from both sides:

$-\frac{3}{2} - 1 = \frac{1}{2}x$

$-\frac{3}{2} - \frac{2}{2} = \frac{1}{2}x$

$-\frac{5}{2} = \frac{1}{2}x$

Now, multiply both sides by 2 to solve for $x$:

$2 \cdot (-\frac{5}{2}) = 2 \cdot (\frac{1}{2}x)$

$-5 = x$

So, $x = -5$.

3. Final Answer

$x = -5$