Let's solve the problem step by step:

We are given a rectangle with the perimeter of 42 inches. The length of the rectangle is $5x$ and the width is $x - 3$.

The formula for the perimeter $P$ of a rectangle is:

$P = 2(\text{Length} + \text{Width})$

Substituting the values we know:

$42 = 2(5x + (x - 3))$

Now let's solve for $x$:

Step 1: Simplify the equation inside the parentheses:

$42 = 2(5x + x - 3)$ $42 = 2(6x - 3)$

Step 2: Distribute the 2 on the right-hand side:

$42 = 12x - 6$

Step 3: Add 6 to both sides to isolate the term with $x$:

$42 + 6 = 12x$ $48 = 12x$

Step 4: Divide both sides by 12:

$x = \frac{48}{12} = 4$

Thus, the value of $x$ is $\boxed{4}$.

Would you like a detailed explanation of any step or have additional questions? Here are 5 related questions to deepen your understanding:

1. How would the equation change if the perimeter were doubled?
2. What would be the perimeter if $x = 6$?
3. How can we find the area of the rectangle using $x = 4$?
4. What is the length and width of the rectangle once $x = 4$?
5. How would the result change if the width formula was $x + 3$ instead of $x - 3$?

Tip: Always check if the perimeter equation is simplified before solving for the variable.