Practice Test? Add. Simplify your answer if you can. $$ \begin{array}{l} \frac{7}{5}+\frac{8}{5}= \ \frac{6}{x+5}+\frac{5}{x+5}= \ \frac{8}{a^{2}}+\frac{1}{a}= \ x+\frac{x}{4}= \ \frac{2}{x+10}+\frac{3}{5 x+50}= \ \frac{12}{x^{2}-9}+\frac{2}{x+3}= \ \frac{x}{x+4}+\frac{2}{x-1}= \end{array} $$ $$ \begin{array}{l} \frac{x}{2 x}+\frac{3}{2 x}= \ \frac{x-2}{x+2}+\frac{x+6}{x+2}= \ \frac{3}{p}+\frac{4}{9}= \ 3+\frac{2}{x-5}= \end{array} $$

Question

UpStudy AI · Accepted Answer

Sure! Let's simplify each of the given addition problems step by step.

### Problem 1
$$
\frac{7}{5} + \frac{8}{5}
$$

**Solution:**
Since the denominators are the same, add the numerators:
$$
\frac{7 + 8}{5} = \frac{15}{5} = 3
$$

**Answer:** 
$$
3
$$

---

### Problem 2
$$
\frac{6}{x + 5} + \frac{5}{x + 5}
$$

**Solution:**
Same denominator, add the numerators:
$$
\frac{6 + 5}{x + 5} = \frac{11}{x + 5}
$$

**Answer:** 
$$
\frac{11}{x + 5}
$$

---

### Problem 3
$$
\frac{8}{a^{2}} + \frac{1}{a}
$$

**Solution:**
Find a common denominator, which is $$ a^2 $$:
$$
\frac{8}{a^2} + \frac{1 \cdot a}{a \cdot a} = \frac{8}{a^2} + \frac{a}{a^2} = \frac{8 + a}{a^2}
$$

**Answer:** 
$$
\frac{a + 8}{a^{2}}
$$

---

### Problem 4
$$
x + \frac{x}{4}
$$

**Solution:**
Rewrite $$ x $$ as $$ \frac{4x}{4} $$ to have a common denominator:
$$
\frac{4x}{4} + \frac{x}{4} = \frac{4x + x}{4} = \frac{5x}{4}
$$

**Answer:** 
$$
\frac{5x}{4}
$$

---

### Problem 5
$$
\frac{2}{x + 10} + \frac{3}{5x + 50}
$$

**Solution:**
Factor the denominator of the second fraction:
$$
5x + 50 = 5(x + 10)
$$
So,
$$
\frac{2}{x + 10} + \frac{3}{5(x + 10)} = \frac{2 \cdot 5}{5(x + 10)} + \frac{3}{5(x + 10)} = \frac{10 + 3}{5(x + 10)} = \frac{13}{5(x + 10)}
$$

**Answer:** 
$$
\frac{13}{5(x + 10)}
$$

---

### Problem 6
$$
\frac{12}{x^{2} - 9} + \frac{2}{x + 3}
$$

**Solution:**
Factor the denominator $$ x^2 - 9 $$:
$$
x^2 - 9 = (x + 3)(x - 3)
$$
Rewrite the expression:
$$
\frac{12}{(x + 3)(x - 3)} + \frac{2}{x + 3}
$$
Find a common denominator $$ (x + 3)(x - 3) $$:
$$
\frac{12}{(x + 3)(x - 3)} + \frac{2(x - 3)}{(x + 3)(x - 3)} = \frac{12 + 2(x - 3)}{(x + 3)(x - 3)} = \frac{12 + 2x - 6}{(x + 3)(x - 3)} = \frac{2x + 6}{(x + 3)(x - 3)} = \frac{2(x + 3)}{(x + 3)(x - 3)} = \frac{2}{x - 3}
$$
*(Assuming $$ x \neq -3 $$ and $$ x \neq 3 $$)*

**Answer:** 
$$
\frac{2}{x - 3}
$$

---

### Problem 7
$$
\frac{x}{x + 4} + \frac{2}{x - 1}
$$

**Solution:**
Find a common denominator $$ (x + 4)(x - 1) $$:
$$
\frac{x(x - 1)}{(x + 4)(x - 1)} + \frac{2(x + 4)}{(x + 4)(x - 1)} = \frac{x(x - 1) + 2(x + 4)}{(x + 4)(x - 1)} = \frac{x^2 - x + 2x + 8}{(x + 4)(x - 1)} = \frac{x^2 + x + 8}{(x + 4)(x - 1)}
$$

**Answer:** 
$$
\frac{x^{2} + x + 8}{(x + 4)(x - 1)}
$$

---

### Problem 8
$$
\frac{x}{2x} + \frac{3}{2x}
$$

**Solution:**
Simplify $$ \frac{x}{2x} $$:
$$
\frac{x}{2x} = \frac{1}{2} \quad (\text{assuming } x \neq 0)
$$
Now add:
$$
\frac{1}{2} + \frac{3}{2x} = \frac{x + 3}{2x}
$$

**Answer:** 
$$
\frac{x + 3}{2x}
$$

---

### Problem 9
$$
\frac{x - 2}{x + 2} + \frac{x + 6}{x + 2}
$$

**Solution:**
Same denominator, add the numerators:
$$
\frac{(x - 2) + (x + 6)}{x + 2} = \frac{2x + 4}{x + 2} = \frac{2(x + 2)}{x + 2} = 2 \quad (\text{assuming } x \neq -2)
$$

**Answer:** 
$$
2
$$

---

### Problem 10
$$
\frac{3}{p} + \frac{4}{9}
$$

**Solution:**
Find a common denominator, which is $$ 9p $$:
$$
\frac{3 \cdot 9}{p \cdot 9} + \frac{4 \cdot p}{9 \cdot p} = \frac{27}{9p} + \frac{4p}{9p} = \frac{27 + 4p}{9p}
$$

**Answer:** 
$$
\frac{4p + 27}{9p}
$$

---

### Problem 11
$$
3 + \frac{2}{x - 5}
$$

**Solution:**
Rewrite $$ 3 $$ with the common denominator $$ x - 5 $$:
$$
3 = \frac{3(x - 5)}{x - 5}
$$
Now add:
$$
\frac{3(x - 5)}{x - 5} + \frac{2}{x - 5} = \frac{3x - 15 + 2}{x - 5} = \frac{3x - 13}{x - 5}
$$

**Answer:** 
$$
\frac{3x - 13}{x - 5}
$$

---

Feel free to ask if you need further explanations or assistance with these problems!
$$ \begin{array}{l} 3 \ \frac{11}{x + 5} \ \frac{a + 8}{a^{2}} \ \frac{5x}{4} \ \frac{13}{5(x + 10)} \ \frac{2}{x - 3} \ \frac{x^{2} + x + 8}{(x + 4)(x - 1)} \ \frac{x + 3}{2x} \ 2 \ \frac{4p + 27}{9p} \ \frac{3x - 13}{x - 5} \end{array} $$

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