https://file.aiquickdraw.com/mathbot/file/1726065301604z3v21cf1.jpg

The equation from the image is:

\[
\frac{h - x}{2h - x} + \frac{h - x}{2 + x} = 1
\]

Let's solve this step by step.

### Step 1: Combine the fractions
To combine the fractions, we need to find a common denominator. The denominators are \(2h - x\) and \(2 + x\), so the common denominator will be the product of these two:

\[
\text{Common denominator} = (2h - x)(2 + x)
\]

Now, rewrite both fractions with the common denominator:

\[
\frac{(h - x)(2 + x)}{(2h - x)(2 + x)} + \frac{(h - x)(2h - x)}{(2h - x)(2 + x)} = 1
\]

### Step 2: Combine the numerators
Since the denominators are now the same, we can combine the numerators:

\[
\frac{(h - x)(2 + x) + (h - x)(2h - x)}{(2h - x)(2 + x)} = 1
\]

### Step 3: Eliminate the denominator
To eliminate the denominator, multiply both sides of the equation by \((2h - x)(2 + x)\):

\[
(h - x)(2 + x) + (h - x)(2h - x) = (2h - x)(2 + x)
\]

### Step 4: Expand both sides
Let's first expand both sides of the equation:

For the left-hand side:

\[
(h - x)(2 + x) = h(2 + x) - x(2 + x) = 2h + hx - 2x - x^2
\]
\[
(h - x)(2h - x) = h(2h - x) - x(2h - x) = 2h^2 - hx - 2hx + x^2 = 2h^2 - 3hx + x^2
\]

So, the left-hand side becomes:

\[
(2h + hx - 2x - x^2) + (2h^2 - 3hx + x^2)
\]

Simplifying:

\[
= 2h + 2h^2 + hx - 3hx - 2x + x^2 - x^2 = 2h + 2h^2 - 2hx - 2x
\]

For the right-hand side:

\[
(2h - x)(2 + x) = 2h(2 + x) - x(2 + x) = 4h + 2hx - 2x - x^2
\]

### Step 5: Set both sides equal and simplify
Now set the left-hand side equal to the right-hand side:

\[
2h + 2h^2 - 2hx - 2x = 4h + 2hx - 2x - x^2
\]

Now, simplify the equation:

\[
2h + 2h^2 - 2hx - 2x = 4h + 2hx - 2x - x^2
\]

Move all terms to one side:

\[
2h + 2h^2 - 2hx - 2x - 4h - 2hx + 2x + x^2 = 0
\]

Simplify further:

\[
2h^2 - 2h - 4hx + x^2 = 0
\]

This is the simplified form of the equation.

Would you like further assistance with solving for \(x\) or \(h\)? Here are 5 related questions to expand on this problem:

1. How do you solve a quadratic equation like the one derived here?
2. Can the equation be solved explicitly for \(h\) or \(x\)?
3. What happens if we try to graph the two expressions to find solutions?
4. How does factoring help simplify such equations?
5. Can we use numerical methods to find solutions if exact algebraic solutions are hard to derive?

**Tip:** When dealing with rational equations, always check for potential restrictions in the domain by setting denominators not equal to zero.

Solve the rational equation: (h - x)/(2h - x) + (h - x)/(2 + x) = 1

Learn how to solve the rational equation (h - x)/(2h - x) + (h - x)/(2 + x) = 1. This step-by-step guide includes finding a common denominator, simplifying expressions, and expanding both sides. Suitable for students in Grades 10-12.

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