https://file.aiquickdraw.com/mathbot/file/17291831436068yazgstp.jpg

Let's solve the equation from the image:

\[
\frac{x + 5}{x - 4} + 1 = \frac{x + 1}{x - 3}
\]

### Step 1: Eliminate fractions by finding the least common denominator (LCD).
The LCD is \((x - 4)(x - 3)\). Multiply both sides of the equation by this LCD to get rid of the denominators:

\[
(x - 4)(x - 3) \cdot \frac{x + 5}{x - 4} + (x - 4)(x - 3) \cdot 1 = (x - 4)(x - 3) \cdot \frac{x + 1}{x - 3}
\]

Simplify:

\[
(x + 5)(x - 3) + (x - 4)(x - 3) = (x + 1)(x - 4)
\]

### Step 2: Expand the terms.
1. Expand \((x + 5)(x - 3)\):

\[
x^2 - 3x + 5x - 15 = x^2 + 2x - 15
\]

2. Expand \((x - 4)(x - 3)\):

\[
x^2 - 3x - 4x + 12 = x^2 - 7x + 12
\]

3. Expand \((x + 1)(x - 4)\):

\[
x^2 - 4x + x - 4 = x^2 - 3x - 4
\]

Now substitute the expanded forms into the equation:

\[
(x^2 + 2x - 15) + (x^2 - 7x + 12) = x^2 - 3x - 4
\]

### Step 3: Combine like terms.
Simplify both sides:

\[
2x^2 - 5x - 3 = x^2 - 3x - 4
\]

Move all terms to one side:

\[
2x^2 - 5x - 3 - x^2 + 3x + 4 = 0
\]

\[
x^2 - 2x + 1 = 0
\]

### Step 4: Solve the quadratic equation.
\[
x^2 - 2x + 1 = 0
\]

This is a perfect square:

\[
(x - 1)^2 = 0
\]

So, the solution is:

\[
x = 1
\]

### Step 5: Check for restrictions.
The original denominators were \(x - 4\) and \(x - 3\). To avoid division by zero:

\[
x \neq 4, \quad x \neq 3
\]

Since \(x = 1\) does not violate these restrictions, the solution is valid.

### Final Answer:
\[
x = 1
\]

Would you like more details or have any other questions?

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**Related questions:**
1. How do we solve equations with more complex rational expressions?
2. What happens if a solution violates the domain restrictions?
3. Can all quadratic equations be solved by factoring?
4. How do we determine whether a quadratic equation has real or complex solutions?
5. How can we verify solutions in rational equations?

**Tip:** Always check for restrictions in rational equations to ensure your solutions are valid.

Solve for x: (x + 5) / (x - 4) + 1 = (x + 1) / (x - 3)

Learn how to solve the rational equation (x + 5)/(x - 4) + 1 = (x + 1)/(x - 3) with a step-by-step guide. This problem focuses on eliminating fractions by multiplying by the least common denominator and solving the resulting quadratic equation. Suitable for students in Grades 9-11.

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