https://file.aiquickdraw.com/mathbot/file/1729182980723aw9jvk1d.jpg

We are given the equation:

\[
\frac{w+6}{w+5} - 1 = \frac{w-1}{w-5}
\]

Let's solve this step by step.

### Step 1: Simplify the equation

First, rewrite \( -1 \) as \( \frac{w+5}{w+5} \) to have a common denominator:

\[
\frac{w+6}{w+5} - \frac{w+5}{w+5} = \frac{w-1}{w-5}
\]

Now, simplify the left-hand side:

\[
\frac{(w+6) - (w+5)}{w+5} = \frac{w-1}{w-5}
\]

This simplifies to:

\[
\frac{w + 6 - w - 5}{w+5} = \frac{w-1}{w-5}
\]

\[
\frac{1}{w+5} = \frac{w-1}{w-5}
\]

### Step 2: Cross-multiply

Now, cross-multiply to eliminate the denominators:

\[
1 \cdot (w - 5) = (w + 5) \cdot (w - 1)
\]

This simplifies to:

\[
w - 5 = (w + 5)(w - 1)
\]

### Step 3: Expand and solve

Expand the right-hand side:

\[
w - 5 = w^2 - w + 5w - 5
\]

Simplifying:

\[
w - 5 = w^2 + 4w - 5
\]

### Step 4: Rearrange into a quadratic equation

Move all terms to one side of the equation:

\[
0 = w^2 + 4w - 5 - w + 5
\]

Simplify further:

\[
0 = w^2 + 3w
\]

### Step 5: Solve for \( w \)

Factor the quadratic:

\[
0 = w(w + 3)
\]

This gives two solutions:

\[
w = 0 \quad \text{or} \quad w = -3
\]

### Step 6: Check for restrictions

The original equation has denominators \( w+5 \) and \( w-5 \). Therefore, \( w \neq -5 \) and \( w \neq 5 \), to avoid division by zero. Neither of our solutions \( w = 0 \) or \( w = -3 \) violate these restrictions.

### Final Answer:
The solutions are \( w = 0 \) and \( w = -3 \).

Would you like more details on any step? Here are some follow-up questions to consider:

1. What is cross-multiplication, and how does it help simplify rational equations?
2. How can you factor quadratic equations effectively?
3. What are the restrictions on solutions when dealing with rational equations?
4. How do you recognize when a quadratic equation has no real solutions?
5. How can you check your solutions when dealing with complex fractions?

**Tip:** Always check for restrictions on the variable when working with equations that involve fractions or rational expressions.

Solve for w: (w + 6) / (w + 5) - 1 = (w - 1) / (w - 5).

Learn how to solve a rational equation leading to a quadratic equation step-by-step. This solution demonstrates cross-multiplication, equation expansion, and factoring, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation