https://file.aiquickdraw.com/mathbot/file/1727046258324li0yk455.jpg

I see the image you've uploaded with a mathematical expression to simplify. Let's focus on the visible expression in the problem:

\[
\frac{z + 2}{z + 3} + \frac{z - 1}{z + 1}
\]

The instruction is to simplify the rational expression without factoring the numerator or denominator. Here’s how to approach it:

### Steps to Simplify:

1. **Find a common denominator** for both fractions:
   - The denominators are \( z + 3 \) and \( z + 1 \). The least common denominator (LCD) is the product of both denominators: \( (z + 3)(z + 1) \).

2. **Rewrite each fraction with the LCD**:
   - First fraction: \(\frac{z + 2}{z + 3} = \frac{(z + 2)(z + 1)}{(z + 3)(z + 1)}\).
   - Second fraction: \(\frac{z - 1}{z + 1} = \frac{(z - 1)(z + 3)}{(z + 3)(z + 1)}\).

3. **Add the numerators**:
   - \[
   \frac{(z + 2)(z + 1) + (z - 1)(z + 3)}{(z + 3)(z + 1)}
   \]

4. **Expand the numerators**:
   - Expand \((z + 2)(z + 1) = z^2 + z + 2z + 2 = z^2 + 3z + 2\).
   - Expand \((z - 1)(z + 3) = z^2 + 3z - z - 3 = z^2 + 2z - 3\).

5. **Combine the expanded terms**:
   - The sum of the numerators is:
   \[
   (z^2 + 3z + 2) + (z^2 + 2z - 3) = 2z^2 + 5z - 1.
   \]

6. **Final simplified expression**:
   - The final expression is:
   \[
   \frac{2z^2 + 5z - 1}{(z + 3)(z + 1)}.
   \]

This is the simplified form of the rational expression.

Would you like more details or have any questions?

---

Here are 5 related questions for further practice:

1. How would you simplify \(\frac{x - 2}{x + 5} + \frac{x + 1}{x + 2}\)?
2. What is the common denominator for the expression \(\frac{y + 3}{y - 1} + \frac{y - 4}{y + 2}\)?
3. Can you factor the numerator of \(\frac{2z^2 + 5z - 1}{(z + 3)(z + 1)}\)?
4. How do you subtract rational expressions with different denominators?
5. What changes if the denominators in the original question were \(z^2 + 1\) and \(z^2 - 4\)?

**Tip:** Always look for the least common denominator when adding or subtracting rational expressions!

Simplify the rational expression: (z + 2) / (z + 3) + (z - 1) / (z + 1)

This guide explains how to simplify the rational expression (z + 2) / (z + 3) + (z - 1) / (z + 1) by finding a common denominator, expanding the numerators, and combining terms. Learn step-by-step how to work through this algebraic problem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation