Let's solve the problem step by step.

Given:

1. The height of the triangle = $4 \, \text{cm}$.
2. The base of the triangle = $8x - 2 \, \text{cm}$.
3. The area of the triangle = $12 \, \text{cm}^2$.

The formula for the area of a triangle is:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}.$

Step 1: Substituting the given values

$12 = \frac{1}{2} \times (8x - 2) \times 4.$

Step 2: Simplify the equation

Multiply both sides by 2 to eliminate the fraction: $24 = (8x - 2) \times 4.$

Distribute the $4$ across $8x - 2$: $24 = 32x - 8.$

Step 3: Solve for $x$

Add $8$ to both sides: $32 = 32x.$

Divide both sides by $32$: $x = 1.$

Final Answer:

The value of $x$ is 1.

Would you like a detailed explanation of any step?

Here are 5 related questions:

1. What is the formula for the area of other geometric shapes (like rectangles or trapeziums)?
2. How do you verify your answer by substituting $x = 1$ back into the problem?
3. What happens to the area if the base increases but the height remains the same?
4. How would the calculation change if the height was unknown but the area and base were given?
5. Can you explain how to solve equations with fractions systematically?

Tip: When solving equations with fractions, always eliminate the fractions early by multiplying through by the denominator.