We are asked to solve the linear equation $4x - 0.3 = 0.5x - 6$ for the variable $x$.

1. Problem Description

We are asked to solve the linear equation $4x - 0.3 = 0.5x - 6$ for the variable $x$.

2. Solution Steps

We want to isolate $x$ on one side of the equation. First, subtract $0.5x$ from both sides:

$4x - 0.3 - 0.5x = 0.5x - 6 - 0.5x$

$3.5x - 0.3 = -6$

Next, add $0.3$ to both sides:

$3.5x - 0.3 + 0.3 = -6 + 0.3$

$3.5x = -5.7$

Finally, divide both sides by $3.5$:

$x = \frac{-5.7}{3.5}$

To eliminate decimals, multiply both the numerator and the denominator by 10:

$x = \frac{-57}{35}$

We can express this as a mixed number if we want:

$-57 \div 35 = -1$ with a remainder of $-22$.

$x = -1 \frac{22}{35}$. As a decimal, $x = -1.62857\ldots$

3. Final Answer

$x = \frac{-57}{35}$