https://file.aiquickdraw.com/mathbot/file/1727227420791ygnn2nvv.jpg

The expression in the image is:

\[
\frac{1}{y + 2} + \frac{9}{y + 1}
\]

To add these two rational expressions, we need to find a common denominator.

### Step 1: Find the Least Common Denominator (LCD)
The denominators are \( y + 2 \) and \( y + 1 \). The least common denominator (LCD) will be the product of these two terms:

\[
\text{LCD} = (y + 2)(y + 1)
\]

### Step 2: Rewrite Each Fraction with the LCD
Now, rewrite each fraction with the LCD as the denominator.

For \( \frac{1}{y + 2} \), multiply both the numerator and the denominator by \( y + 1 \):

\[
\frac{1}{y + 2} = \frac{1 \cdot (y + 1)}{(y + 2)(y + 1)} = \frac{y + 1}{(y + 2)(y + 1)}
\]

For \( \frac{9}{y + 1} \), multiply both the numerator and the denominator by \( y + 2 \):

\[
\frac{9}{y + 1} = \frac{9 \cdot (y + 2)}{(y + 1)(y + 2)} = \frac{9(y + 2)}{(y + 1)(y + 2)}
\]

### Step 3: Add the Fractions
Now, add the two fractions:

\[
\frac{y + 1}{(y + 2)(y + 1)} + \frac{9(y + 2)}{(y + 1)(y + 2)} = \frac{(y + 1) + 9(y + 2)}{(y + 1)(y + 2)}
\]

### Step 4: Simplify the Numerator
Simplify the numerator:

\[
(y + 1) + 9(y + 2) = y + 1 + 9y + 18 = 10y + 19
\]

So, the expression becomes:

\[
\frac{10y + 19}{(y + 1)(y + 2)}
\]

### Final Answer:
The simplified form of the given expression is:

\[
\frac{10y + 19}{(y + 1)(y + 2)}
\]

Let me know if you'd like further explanation or details!

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Here are 5 related questions to expand on this concept:
1. How would you simplify an expression with three fractions with different denominators?
2. What happens if one of the terms in the denominator contains a square, like \( y^2 + 1 \)?
3. How would you factor and simplify a rational expression like \( \frac{y^2 + 3y + 2}{y^2 - 1} \)?
4. How do you handle complex rational expressions where both the numerator and denominator are fractions?
5. Can you explain how to perform polynomial long division when simplifying rational expressions?

**Tip**: Always factor denominators when possible to find the least common denominator more easily!

Simplify the following rational expression: \( \frac{1}{y + 2} + \frac{9}{y + 1} \).

Learn how to simplify the rational expression 1/(y+2) + 9/(y+1) with a step-by-step solution. This problem involves key algebraic concepts such as finding the least common denominator and simplifying rational expressions. Suitable for students in Grades 8-10.

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