Let's solve both equations step by step from the image.

Equation 1:

$\frac{x}{2} - \frac{x}{4} = \frac{3}{4}$

Step 1: Simplify the left-hand side.

We need a common denominator for $\frac{x}{2}$ and $\frac{x}{4}$. The common denominator is 4.

$\frac{x}{2} = \frac{2x}{4}$

Now substitute into the equation:

$\frac{2x}{4} - \frac{x}{4} = \frac{3}{4}$

Simplify the left-hand side:

$\frac{2x - x}{4} = \frac{3}{4}$

$\frac{x}{4} = \frac{3}{4}$

Step 2: Solve for $x$.

Since the denominators are the same, we can equate the numerators:

$x = 3$

Check:

Substitute $x = 3$ into the original equation:

$\frac{3}{2} - \frac{3}{4} = \frac{3}{4}$

Convert $\frac{3}{2}$ to a fraction with denominator 4:

$\frac{6}{4} - \frac{3}{4} = \frac{3}{4}$

$\frac{3}{4} = \frac{3}{4}$

So, $x = 3$ is correct.

Equation 2:

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

Step 1: Subtract 1 from both sides.

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

Rewrite 1 with the denominator $x - 2$:

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

Simplify the left-hand side:

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

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

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

Step 2: Cross multiply.

$2(x - 3) = (x - 2)$

Expand both sides:

$2x - 6 = x - 2$

Step 3: Solve for $x$.

Subtract $x$ from both sides:

$x - 6 = -2$

Add 6 to both sides:

$x = 4$

Check:

Substitute $x = 4$ into the original equation:

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

Simplify both sides:

$\frac{4}{2} = 1 + \frac{1}{1}$

$2 = 1 + 1$

$2 = 2$

So, $x = 4$ is correct.

Would you like more details on any of the steps? Here are some related questions to deepen your understanding:

1. How do you find a common denominator for fractions?
2. What is the significance of cross-multiplying when solving rational equations?
3. How would the solution change if the constants on the right-hand side of both equations were different?
4. What are some potential pitfalls when solving rational equations?
5. Can you create a new equation similar to the ones provided and solve it?

Tip: Always verify solutions by substituting the variable back into the original equation to ensure the solution is correct.