Algebraic Fractions worksheet with eight problems to simplify, designed for practice in combining and reducing algebraic fractions.

To solve the given problems involving algebraic fractions, we need to simplify each expression step by step. Let's go through each problem systematically.

---

Problem 1: Simplify \( \frac{3x}{7} + \frac{x+3}{7} \)

Since the denominators are the same, we can combine the numerators directly:
\[
\frac{3x}{7} + \frac{x+3}{7} = \frac{3x + (x + 3)}{7}
\]
Simplify the numerator:
\[
3x + x + 3 = 4x + 3
\]
So the expression becomes:
\[
\frac{4x + 3}{7}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{4x + 3}{7}}
\]

---

Problem 2: Simplify \( \frac{8x}{3} + \frac{x-3}{12} \)

The denominators are different, so we need a common denominator. The least common multiple (LCM) of 3 and 12 is 12. Rewrite each fraction with the common denominator:
\[
\frac{8x}{3} = \frac{8x \cdot 4}{3 \cdot 4} = \frac{32x}{12}
\]
\[
\frac{x-3}{12} \text{ remains as it is.}
\]
Now add the fractions:
\[
\frac{32x}{12} + \frac{x-3}{12} = \frac{32x + (x - 3)}{12}
\]
Simplify the numerator:
\[
32x + x - 3 = 33x - 3
\]
So the expression becomes:
\[
\frac{33x - 3}{12}
\]
Factor out the greatest common divisor (GCD) of the numerator and denominator:
\[
\frac{33x - 3}{12} = \frac{3(11x - 1)}{12} = \frac{11x - 1}{4}
\]

Answer:
\[
\boxed{\frac{11x - 1}{4}}
\]

---

Problem 3: Simplify \( \frac{9x}{8} - \frac{x-5}{16} \)

The denominators are different, so we need a common denominator. The LCM of 8 and 16 is 16. Rewrite each fraction with the common denominator:
\[
\frac{9x}{8} = \frac{9x \cdot 2}{8 \cdot 2} = \frac{18x}{16}
\]
\[
\frac{x-5}{16} \text{ remains as it is.}
\]
Now subtract the fractions:
\[
\frac{18x}{16} - \frac{x-5}{16} = \frac{18x - (x - 5)}{16}
\]
Simplify the numerator:
\[
18x - x + 5 = 17x + 5
\]
So the expression becomes:
\[
\frac{17x + 5}{16}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{17x + 5}{16}}
\]

---

Problem 4: Simplify \( \frac{x+7}{3} + \frac{3x-2}{18} \)

The denominators are different, so we need a common denominator. The LCM of 3 and 18 is 18. Rewrite each fraction with the common denominator:
\[
\frac{x+7}{3} = \frac{(x+7) \cdot 6}{3 \cdot 6} = \frac{6(x+7)}{18} = \frac{6x + 42}{18}
\]
\[
\frac{3x-2}{18} \text{ remains as it is.}
\]
Now add the fractions:
\[
\frac{6x + 42}{18} + \frac{3x - 2}{18} = \frac{(6x + 42) + (3x - 2)}{18}
\]
Simplify the numerator:
\[
6x + 3x + 42 - 2 = 9x + 40
\]
So the expression becomes:
\[
\frac{9x + 40}{18}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{9x + 40}{18}}
\]

---

Problem 5: Simplify \( \frac{7x}{8} + \frac{3x}{10} - \frac{x}{5} \)

The denominators are different, so we need a common denominator. The LCM of 8, 10, and 5 is 40. Rewrite each fraction with the common denominator:
\[
\frac{7x}{8} = \frac{7x \cdot 5}{8 \cdot 5} = \frac{35x}{40}
\]
\[
\frac{3x}{10} = \frac{3x \cdot 4}{10 \cdot 4} = \frac{12x}{40}
\]
\[
\frac{x}{5} = \frac{x \cdot 8}{5 \cdot 8} = \frac{8x}{40}
\]
Now combine the fractions:
\[
\frac{35x}{40} + \frac{12x}{40} - \frac{8x}{40} = \frac{35x + 12x - 8x}{40}
\]
Simplify the numerator:
\[
35x + 12x - 8x = 39x
\]
So the expression becomes:
\[
\frac{39x}{40}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{39x}{40}}
\]

---

Problem 6: Simplify \( \frac{2x}{11} - \frac{5(x-2)}{11} \)

The denominators are the same, so we can combine the numerators directly:
\[
\frac{2x}{11} - \frac{5(x-2)}{11} = \frac{2x - 5(x-2)}{11}
\]
Distribute the 5 in the second term:
\[
2x - 5(x-2) = 2x - 5x + 10 = -3x + 10
\]
So the expression becomes:
\[
\frac{-3x + 10}{11}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{-3x + 10}{11}}
\]

---

Problem 7: Simplify \( \frac{2}{x} - \frac{3}{x} + \frac{7}{2x} \)

The denominators are different, so we need a common denominator. The LCM of \(x\) and \(2x\) is \(2x\). Rewrite each fraction with the common denominator:
\[
\frac{2}{x} = \frac{2 \cdot 2}{x \cdot 2} = \frac{4}{2x}
\]
\[
\frac{3}{x} = \frac{3 \cdot 2}{x \cdot 2} = \frac{6}{2x}
\]
\[
\frac{7}{2x} \text{ remains as it is.}
\]
Now combine the fractions:
\[
\frac{4}{2x} - \frac{6}{2x} + \frac{7}{2x} = \frac{4 - 6 + 7}{2x}
\]
Simplify the numerator:
\[
4 - 6 + 7 = 5
\]
So the expression becomes:
\[
\frac{5}{2x}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{5}{2x}}
\]

---

Problem 8: Simplify \( \frac{4x}{7} - \frac{x-3}{8} \)

The denominators are different, so we need a common denominator. The LCM of 7 and 8 is 56. Rewrite each fraction with the common denominator:
\[
\frac{4x}{7} = \frac{4x \cdot 8}{7 \cdot 8} = \frac{32x}{56}
\]
\[
\frac{x-3}{8} = \frac{(x-3) \cdot 7}{8 \cdot 7} = \frac{7(x-3)}{56} = \frac{7x - 21}{56}
\]
Now subtract the fractions:
\[
\frac{32x}{56} - \frac{7x - 21}{56} = \frac{32x - (7x - 21)}{56}
\]
Simplify the numerator:
\[
32x - 7x + 21 = 25x + 21
\]
So the expression becomes:
\[
\frac{25x + 21}{56}
\]
This is already in its simplest form.

Answer:
\[
\boxed{\frac{25x + 21}{56}}
\]

---

Final Answers:

1. \(\boxed{\frac{4x + 3}{7}}\)
2. \(\boxed{\frac{11x - 1}{4}}\)
3. \(\boxed{\frac{17x + 5}{16}}\)
4. \(\boxed{\frac{9x + 40}{18}}\)
5. \(\boxed{\frac{39x}{40}}\)
6. \(\boxed{\frac{-3x + 10}{11}}\)
7. \(\boxed{\frac{5}{2x}}\)
8. \(\boxed{\frac{25x + 21}{56}}\)