addition and subtraction of rational expressions with different denominators involve converting each expression into equivalent rational expressions with the least common denominator.
$\frac{3}{x - 5} + \frac{x}{x + 2}$
$\frac{m}{m + 3} - \frac{2}{m - 1}$
$\frac{4 - 2p}{p^2 - p - 2} + \frac{p + 2}{p + 1}$
$\frac{4 - a}{a - 3} - \frac{a - 3}{a^2 - 6a + 9}$
$\frac{3}{u^2 + 7u + 10} - \frac{2}{u^2 + 8u + 15}$
$\frac{1}{n^2 - 1} + \frac{3}{n^2 - 4n + 3}$
$\frac{8}{y + 9} - \frac{9}{3y + 27}$
$\frac{1}{b + 1} + \frac{2}{b^2 - 1}$
$\frac{10}{x - 5} - \frac{2x^2 + 4x}{x^2 - 3x - 10}$

Question

Accepted Answer

# Explanation:
---
### 1. $\boldsymbol{\frac{3}{x-5} + \frac{x}{x+2}}$
## Step1: Find common denominator
Least common denominator is $(x-5)(x+2)$
## Step2: Rewrite each fraction
$\frac{3(x+2)}{(x-5)(x+2)} + \frac{x(x-5)}{(x-5)(x+2)}$
## Step3: Combine numerators
$\frac{3(x+2) + x(x-5)}{(x-5)(x+2)}$
## Step4: Expand and simplify numerator
$\frac{3x+6 + x^2 -5x}{(x-5)(x+2)} = \frac{x^2 -2x +6}{(x-5)(x+2)}$

---
### 2. $\boldsymbol{\frac{m}{m+3} - \frac{2}{m-1}}$
## Step1: Find common denominator
Least common denominator is $(m+3)(m-1)$
## Step2: Rewrite each fraction
$\frac{m(m-1)}{(m+3)(m-1)} - \frac{2(m+3)}{(m+3)(m-1)}$
## Step3: Combine numerators
$\frac{m(m-1) - 2(m+3)}{(m+3)(m-1)}$
## Step4: Expand and simplify numerator
$\frac{m^2 -m -2m -6}{(m+3)(m-1)} = \frac{m^2 -3m -6}{(m+3)(m-1)}$

---
### 3. $\boldsymbol{\frac{4-2p}{p^2-p-2} + \frac{p+2}{p+1}}$
## Step1: Factor denominator
$p^2-p-2=(p-2)(p+1)$
## Step2: Find common denominator
Least common denominator is $(p-2)(p+1)$
## Step3: Rewrite second fraction
$\frac{4-2p}{(p-2)(p+1)} + \frac{(p+2)(p-2)}{(p-2)(p+1)}$
## Step4: Factor $4-2p$
$\frac{-2(p-2)}{(p-2)(p+1)} + \frac{(p+2)(p-2)}{(p-2)(p+1)}$
## Step5: Combine numerators
$\frac{-2(p-2) + (p+2)(p-2)}{(p-2)(p+1)}$
## Step6: Factor numerator, cancel terms
$\frac{(p-2)(-2 + p + 2)}{(p-2)(p+1)} = \frac{p}{p+1}$ (where $p
eq 2$)

---
### 4. $\boldsymbol{\frac{4-a}{a-3} - \frac{a-3}{a^2-6a+9}}$
## Step1: Factor denominator
$a^2-6a+9=(a-3)^2$
## Step2: Find common denominator
Least common denominator is $(a-3)^2$
## Step3: Rewrite first fraction
$\frac{(4-a)(a-3)}{(a-3)^2} - \frac{a-3}{(a-3)^2}$
## Step4: Combine numerators
$\frac{(4-a)(a-3) - (a-3)}{(a-3)^2}$
## Step5: Factor numerator, cancel terms
$\frac{(a-3)(4-a -1)}{(a-3)^2} = \frac{3-a}{a-3} = -1$ (where $a
eq 3$)

---
### 5. $\boldsymbol{\frac{3}{u^2+7u+10} - \frac{2}{u^2+8u+15}}$
## Step1: Factor denominators
$u^2+7u+10=(u+2)(u+5)$; $u^2+8u+15=(u+3)(u+5)$
## Step2: Find common denominator
Least common denominator is $(u+2)(u+3)(u+5)$
## Step3: Rewrite each fraction
$\frac{3(u+3)}{(u+2)(u+3)(u+5)} - \frac{2(u+2)}{(u+2)(u+3)(u+5)}$
## Step4: Combine numerators
$\frac{3(u+3) - 2(u+2)}{(u+2)(u+3)(u+5)}$
## Step5: Simplify numerator
$\frac{3u+9-2u-4}{(u+2)(u+3)(u+5)} = \frac{u+5}{(u+2)(u+3)(u+5)} = \frac{1}{(u+2)(u+3)}$ (where $u
eq -5$)

---
### 6. $\boldsymbol{\frac{1}{n^2-1} + \frac{3}{n^2-4n+3}}$
## Step1: Factor denominators
$n^2-1=(n-1)(n+1)$; $n^2-4n+3=(n-1)(n-3)$
## Step2: Find common denominator
Least common denominator is $(n-1)(n+1)(n-3)$
## Step3: Rewrite each fraction
$\frac{n-3}{(n-1)(n+1)(n-3)} + \frac{3(n+1)}{(n-1)(n+1)(n-3)}$
## Step4: Combine numerators
$\frac{n-3 + 3(n+1)}{(n-1)(n+1)(n-3)}$
## Step5: Simplify numerator
$\frac{n-3+3n+3}{(n-1)(n+1)(n-3)} = \frac{4n}{(n-1)(n+1)(n-3)}$

---
### 7. $\boldsymbol{\frac{8}{y+9} - \frac{9}{3y+27}}$
## Step1: Factor second denominator
$3y+27=3(y+9)$
## Step2: Find common denominator
Least common denominator is $3(y+9)$
## Step3: Rewrite first fraction
$\frac{8 \cdot 3}{3(y+9)} - \frac{9}{3(y+9)}$
## Step4: Combine numerators
$\frac{24 - 9}{3(y+9)} = \frac{15}{3(y+9)} = \frac{5}{y+9}$

---
### 8. $\boldsymbol{\frac{1}{b+1} + \frac{2}{b^2-1}}$
## Step1: Factor second denominator
$b^2-1=(b-1)(b+1)$
## Step2: Find common denominator
Least common denominator is $(b-1)(b+1)$
## Step3: Rewrite first fraction
$\frac{b-1}{(b-1)(b+1)} + \frac{2}{(b-1)(b+1)}$
## Step4: Combine numerators
$\frac{b-1 + 2}{(b-1)(b+1)} = \frac{b+1}{(b-1)(b+1)} = \frac{1}{b-1}$ (where $b
eq -1$)

---
### 9. $\boldsymbol{\frac{10}{x-5} - \frac{2x^2+4x}{x^2-3x-10}}$
## Step1: Factor second denominator
$x^2-3x-10=(x-5)(x+2)$
## Step2: Find common denominator
Least common denominator is $(x-5)(x+2)$
## Step3: Rewrite first fraction
$\frac{10(x+2)}{(x-5)(x+2)} - \frac{2x^2+4x}{(x-5)(x+2)}$
## Step4: Factor numerator of second term
$\frac{10(x+2)}{(x-5)(x+2)} - \frac{2x(x+2)}{(x-5)(x+2)}$
## Step5: Combine numerators
$\frac{10(x+2) - 2x(x+2)}{(x-5)(x+2)}$
## Step6: Factor numerator, cancel terms
$\frac{(x+2)(10-2x)}{(x-5)(x+2)} = \frac{-2(x-5)}{x-5} = -2$ (where $x
eq -2, 5$)

---
# Answer:
1. $\boldsymbol{\frac{x^2 -2x +6}{(x-5)(x+2)}}$
2. $\boldsymbol{\frac{m^2 -3m -6}{(m+3)(m-1)}}$
3. $\boldsymbol{\frac{p}{p+1}}$ (for $p
eq 2$)
4. $\boldsymbol{-1}$ (for $a
eq 3$)
5. $\boldsymbol{\frac{1}{(u+2)(u+3)}}$ (for $u
eq -5$)
6. $\boldsymbol{\frac{4n}{(n-1)(n+1)(n-3)}}$
7. $\boldsymbol{\frac{5}{y+9}}$
8. $\boldsymbol{\frac{1}{b-1}}$ (for $b
eq -1$)
9. $\boldsymbol{-2}$ (for $x
eq -2, 5$)

QUESTION IMAGE

Question

Explanation:

1. $\boldsymbol{\frac{3}{x-5} + \frac{x}{x+2}}$

Step1: Find common denominator

Step2: Rewrite each fraction

Step3: Combine numerators

Step4: Expand and simplify numerator

2. $\boldsymbol{\frac{m}{m+3} - \frac{2}{m-1}}$

Step1: Find common denominator

Step2: Rewrite each fraction

Step3: Combine numerators

Step4: Expand and simplify numerator

3. $\boldsymbol{\frac{4-2p}{p^2-p-2} + \frac{p+2}{p+1}}$

Step1: Factor denominator

Step2: Find common denominator

Step3: Rewrite second fraction

Step4: Factor $4-2p$

Step5: Combine numerators

Step6: Factor numerator, cancel terms

4. $\boldsymbol{\frac{4-a}{a-3} - \frac{a-3}{a^2-6a+9}}$

Step1: Factor denominator

Step2: Find common denominator

Step3: Rewrite first fraction

Step4: Combine numerators

Step5: Factor numerator, cancel terms

5. $\boldsymbol{\frac{3}{u^2+7u+10} - \frac{2}{u^2+8u+15}}$

Step1: Factor denominators

Step2: Find common denominator

Step3: Rewrite each fraction

Step4: Combine numerators

Step5: Simplify numerator

6. $\boldsymbol{\frac{1}{n^2-1} + \frac{3}{n^2-4n+3}}$

Step1: Factor denominators

Step2: Find common denominator

Step3: Rewrite each fraction

Step4: Combine numerators

Step5: Simplify numerator

7. $\boldsymbol{\frac{8}{y+9} - \frac{9}{3y+27}}$

Step1: Factor second denominator

Step2: Find common denominator

Step3: Rewrite first fraction

Step4: Combine numerators

8. $\boldsymbol{\frac{1}{b+1} + \frac{2}{b^2-1}}$

Step1: Factor second denominator

Step2: Find common denominator

Step3: Rewrite first fraction

Step4: Combine numerators

9. $\boldsymbol{\frac{10}{x-5} - \frac{2x^2+4x}{x^2-3x-1…

Answer: