https://file.aiquickdraw.com/mathbot/file/1725777241407a6fqsuu9.jpg

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.

Solve for x in the two equations:
1. (x/2) - (x/4) = 3/4
2. (x/(x-2)) = 1 + (1/(x-3))

Learn how to solve two rational equations involving fractions step-by-step. This example includes simplifying fractions, finding a common denominator, and applying cross-multiplication. Suitable for students in Grades 9-10.

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