Precalculus worksheet focusing on fraction operations and algebraic expression simplification.

Problem: Solve the given fractions worksheet problems and explain the solutions.

#### Part 1: Evaluate each expression

1. $ \left(-\frac{1}{6}\right) - \frac{2}{5} $
- To subtract these fractions, find a common denominator. The least common denominator (LCD) of 6 and 5 is 30.
- Convert each fraction:
$$
-\frac{1}{6} = -\frac{5}{30}, \quad \frac{2}{5} = \frac{12}{30}
$$
- Subtract:
$$
-\frac{5}{30} - \frac{12}{30} = -\frac{17}{30}
$$
- Answer: $ -\frac{17}{30} $

2. $ \left(-2\frac{1}{2}\right) + 2\frac{2}{3} $
- Convert mixed numbers to improper fractions:
$$
-2\frac{1}{2} = -\frac{5}{2}, \quad 2\frac{2}{3} = \frac{8}{3}
$$
- Find the LCD of 2 and 3, which is 6. Convert each fraction:
$$
-\frac{5}{2} = -\frac{15}{6}, \quad \frac{8}{3} = \frac{16}{6}
$$
- Add:
$$
-\frac{15}{6} + \frac{16}{6} = \frac{1}{6}
$$
- Answer: $ \frac{1}{6} $

3. $ \left(-\frac{7}{4}\right) - 1\frac{2}{3} $
- Convert the mixed number to an improper fraction:
$$
1\frac{2}{3} = \frac{5}{3}
$$
- Find the LCD of 4 and 3, which is 12. Convert each fraction:
$$
-\frac{7}{4} = -\frac{21}{12}, \quad \frac{5}{3} = \frac{20}{12}
$$
- Subtract:
$$
-\frac{21}{12} - \frac{20}{12} = -\frac{41}{12}
$$
- Answer: $ -\frac{41}{12} $

4. $ \left(-\frac{1}{4}\right) - \left(-\frac{3}{8}\right) $
- Simplify the subtraction of a negative:
$$
-\frac{1}{4} - \left(-\frac{3}{8}\right) = -\frac{1}{4} + \frac{3}{8}
$$
- Find the LCD of 4 and 8, which is 8. Convert each fraction:
$$
-\frac{1}{4} = -\frac{2}{8}, \quad \frac{3}{8} = \frac{3}{8}
$$
- Add:
$$
-\frac{2}{8} + \frac{3}{8} = \frac{1}{8}
$$
- Answer: $ \frac{1}{8} $

5. $ \frac{13}{8} - 1\frac{1}{3} $
- Convert the mixed number to an improper fraction:
$$
1\frac{1}{3} = \frac{4}{3}
$$
- Find the LCD of 8 and 3, which is 24. Convert each fraction:
$$
\frac{13}{8} = \frac{39}{24}, \quad \frac{4}{3} = \frac{32}{24}
$$
- Subtract:
$$
\frac{39}{24} - \frac{32}{24} = \frac{7}{24}
$$
- Answer: $ \frac{7}{24} $

6. $ \left(-\frac{1}{2}\right) + \left(-3\frac{7}{8}\right) $
- Convert the mixed number to an improper fraction:
$$
-3\frac{7}{8} = -\frac{31}{8}
$$
- Find the LCD of 2 and 8, which is 8. Convert each fraction:
$$
-\frac{1}{2} = -\frac{4}{8}, \quad -\frac{31}{8} = -\frac{31}{8}
$$
- Add:
$$
-\frac{4}{8} + \left(-\frac{31}{8}\right) = -\frac{35}{8}
$$
- Answer: $ -\frac{35}{8} $

7. $ \frac{5}{7} + \left(\frac{9}{2}\right) $
- Find the LCD of 7 and 2, which is 14. Convert each fraction:
$$
\frac{5}{7} = \frac{10}{14}, \quad \frac{9}{2} = \frac{63}{14}
$$
- Add:
$$
\frac{10}{14} + \frac{63}{14} = \frac{73}{14}
$$
- Answer: $ \frac{73}{14} $

8. $ (-2) + 3\frac{1}{2} $
- Convert the mixed number to an improper fraction:
$$
3\frac{1}{2} = \frac{7}{2}
$$
- Write -2 as a fraction with a denominator of 2:
$$
-2 = -\frac{4}{2}
$$
- Add:
$$
-\frac{4}{2} + \frac{7}{2} = \frac{3}{2}
$$
- Answer: $ \frac{3}{2} $

9. $ (-2) - \frac{3}{2} $
- Write -2 as a fraction with a denominator of 2:
$$
-2 = -\frac{4}{2}
$$
- Subtract:
$$
-\frac{4}{2} - \frac{3}{2} = -\frac{7}{2}
$$
- Answer: $ -\frac{7}{2} $

10. $ (-1) - \left(-3\frac{1}{4}\right) $
- Simplify the subtraction of a negative:
$$
-1 - \left(-3\frac{1}{4}\right) = -1 + 3\frac{1}{4}
$$
- Convert the mixed number to an improper fraction:
$$
3\frac{1}{4} = \frac{13}{4}
$$
- Write -1 as a fraction with a denominator of 4:
$$
-1 = -\frac{4}{4}
$$
- Add:
$$
-\frac{4}{4} + \frac{13}{4} = \frac{9}{4}
$$
- Answer: $ \frac{9}{4} $

11. $ \left(-2\frac{5}{8}\right) + 3\frac{1}{2} - \frac{3}{4} $
- Convert mixed numbers to improper fractions:
$$
-2\frac{5}{8} = -\frac{21}{8}, \quad 3\frac{1}{2} = \frac{7}{2}
$$
- Find the LCD of 8, 2, and 4, which is 8. Convert each fraction:
$$
-\frac{21}{8}, \quad \frac{7}{2} = \frac{28}{8}, \quad \frac{3}{4} = \frac{6}{8}
$$
- Perform the operations:
$$
-\frac{21}{8} + \frac{28}{8} - \frac{6}{8} = \frac{-21 + 28 - 6}{8} = \frac{1}{8}
$$
- Answer: $ \frac{1}{8} $

12. $ \frac{3}{4} - \frac{7}{6} - 2 $
- Find the LCD of 4, 6, and 1 (for the integer 2), which is 12. Convert each term:
$$
\frac{3}{4} = \frac{9}{12}, \quad \frac{7}{6} = \frac{14}{12}, \quad 2 = \frac{24}{12}
$$
- Perform the operations:
$$
\frac{9}{12} - \frac{14}{12} - \frac{24}{12} = \frac{9 - 14 - 24}{12} = \frac{-29}{12}
$$
- Answer: $ -\frac{29}{12} $

13. $ \left(\frac{5}{4}\right) + \frac{2}{3} + 3 - 1\frac{6}{7} $
- Convert the mixed number to an improper fraction:
$$
1\frac{6}{7} = \frac{13}{7}
$$
- Find the LCD of 4, 3, 1, and 7, which is 84. Convert each term:
$$
\frac{5}{4} = \frac{105}{84}, \quad \frac{2}{3} = \frac{56}{84}, \quad 3 = \frac{252}{84}, \quad \frac{13}{7} = \frac{156}{84}
$$
- Perform the operations:
$$
\frac{105}{84} + \frac{56}{84} + \frac{252}{84} - \frac{156}{84} = \frac{105 + 56 + 252 - 156}{84} = \frac{257}{84}
$$
- Answer: $ \frac{257}{84} $

14. $ \left(-\frac{3}{2}\right) - 4\frac{7}{8} + (-2) - 4\frac{2}{3} $
- Convert mixed numbers to improper fractions:
$$
4\frac{7}{8} = \frac{39}{8}, \quad 4\frac{2}{3} = \frac{14}{3}
$$
- Find the LCD of 2, 8, 1, and 3, which is 24. Convert each term:
$$
-\frac{3}{2} = -\frac{36}{24}, \quad \frac{39}{8} = \frac{117}{24}, \quad -2 = -\frac{48}{24}, \quad \frac{14}{3} = \frac{112}{24}
$$
- Perform the operations:
$$
-\frac{36}{24} - \frac{117}{24} - \frac{48}{24} - \frac{112}{24} = \frac{-36 - 117 - 48 - 112}{24} = \frac{-313}{24}
$$
- Answer: $ -\frac{313}{24} $

15. $ 1 + \frac{9}{5} + 3\frac{1}{6} + \left(-3\frac{1}{7}\right) $
- Convert mixed numbers to improper fractions:
$$
3\frac{1}{6} = \frac{19}{6}, \quad -3\frac{1}{7} = -\frac{22}{7}
$$
- Find the LCD of 1, 5, 6, and 7, which is 210. Convert each term:
$$
1 = \frac{210}{210}, \quad \frac{9}{5} = \frac{378}{210}, \quad \frac{19}{6} = \frac{665}{210}, \quad -\frac{22}{7} = -\frac{660}{210}
$$
- Perform the operations:
$$
\frac{210}{210} + \frac{378}{210} + \frac{665}{210} - \frac{660}{210} = \frac{210 + 378 + 665 - 660}{210} = \frac{593}{210}
$$
- Answer: $ \frac{593}{210} $

16. $ \left(-\frac{5}{3}\right) - \left(-\frac{1}{3}\right) + 2 - \left(-\frac{9}{7}\right) $
- Simplify the subtraction of negatives:
$$
-\frac{5}{3} - \left(-\frac{1}{3}\right) = -\frac{5}{3} + \frac{1}{3}, \quad 2 - \left(-\frac{9}{7}\right) = 2 + \frac{9}{7}
$$
- Combine terms:
$$
-\frac{5}{3} + \frac{1}{3} = -\frac{4}{3}, \quad 2 + \frac{9}{7} = \frac{14}{7} + \frac{9}{7} = \frac{23}{7}
$$
- Find the LCD of 3 and 7, which is 21. Convert each term:
$$
-\frac{4}{3} = -\frac{28}{21}, \quad \frac{23}{7} = \frac{69}{21}
$$
- Add:
$$
-\frac{28}{21} + \frac{69}{21} = \frac{41}{21}
$$
- Answer: $ \frac{41}{21} $

#### Part 2: Simplify each expression

17. $ \left(-\frac{3}{2}a^2 - 1\right) + \left(\frac{5}{2}a^2 + \frac{4}{3}\right) $
- Combine like terms:
$$
-\frac{3}{2}a^2 + \frac{5}{2}a^2 - 1 + \frac{4}{3}
$$
- Simplify the $a^2$ terms:
$$
-\frac{3}{2}a^2 + \frac{5}{2}a^2 = \frac{2}{2}a^2 = a^2
$$
- Simplify the constant terms. Find the LCD of 1 and 3, which is 3:
$$
-1 = -\frac{3}{3}, \quad \frac{4}{3} = \frac{4}{3}
$$
$$
-\frac{3}{3} + \frac{4}{3} = \frac{1}{3}
$$
- Combine:
$$
a^2 + \frac{1}{3}
$$
- Answer: $ a^2 + \frac{1}{3} $

18. $ \left(\frac{14}{3}n - \frac{1}{2}\right) + \left(\frac{7}{4}n + \frac{4}{3}\right) $
- Combine like terms:
$$
\frac{14}{3}n + \frac{7}{4}n - \frac{1}{2} + \frac{4}{3}
$$
- Simplify the $n$ terms. Find the LCD of 3 and 4, which is 12:
$$
\frac{14}{3}n = \frac{56}{12}n, \quad \frac{7}{4}n = \frac{21}{12}n
$$
$$
\frac{56}{12}n + \frac{21}{12}n = \frac{77}{12}n
$$
- Simplify the constant terms. Find the LCD of 2 and 3, which is 6:
$$
-\frac{1}{2} = -\frac{3}{6}, \quad \frac{4}{3} = \frac{8}{6}
$$
$$
-\frac{3}{6} + \frac{8}{6} = \frac{5}{6}
$$
- Combine:
$$
\frac{77}{12}n + \frac{5}{6}
$$
- Answer: $ \frac{77}{12}n + \frac{5}{6} $

19. $ \left(\frac{13}{6}x^4 + \frac{5}{3}\right) + \left(\frac{7}{2} + \frac{5}{2}x^4\right) $
- Combine like terms:
$$
\frac{13}{6}x^4 + \frac{5}{2}x^4 + \frac{5}{3} + \frac{7}{2}
$$
- Simplify the $x^4$ terms. Find the LCD of 6 and 2, which is 6:
$$
\frac{13}{6}x^4 + \frac{5}{2}x^4 = \frac{13}{6}x^4 + \frac{15}{6}x^4 = \frac{28}{6}x^4 = \frac{14}{3}x^4
$$
- Simplify the constant terms. Find the LCD of 3 and 2, which is 6:
$$
\frac{5}{3} = \frac{10}{6}, \quad \frac{7}{2} = \frac{21}{6}
$$
$$
\frac{10}{6} + \frac{21}{6} = \frac{31}{6}
$$
- Combine:
$$
\frac{14}{3}x^4 + \frac{31}{6}
$$
- Answer: $ \frac{14}{3}x^4 + \frac{31}{6} $

20. $ \left(-\frac{19}{5}x^4 + x^2\right) - \left(\frac{2}{3}x^4 - \frac{25}{8}x^2\right) $
- Distribute the negative sign:
$$
-\frac{19}{5}x^4 + x^2 - \frac{2}{3}x^4 + \frac{25}{8}x^2
$$
- Combine like terms:
$$
-\frac{19}{5}x^4 - \frac{2}{3}x^4 + x^2 + \frac{25}{8}x^2
$$
- Simplify the $x^4$ terms. Find the LCD of 5 and 3, which is 15:
$$
-\frac{19}{5}x^4 = -\frac{57}{15}x^4, \quad -\frac{2}{3}x^4 = -\frac{10}{15}x^4
$$
$$
-\frac{57}{15}x^4 - \frac{10}{15}x^4 = -\frac{67}{15}x^4
$$
- Simplify the $x^2$ terms. Find the LCD of 1 and 8, which is 8:
$$
x^2 = \frac{8}{8}x^2, \quad \frac{25}{8}x^2 = \frac{25}{8}x^2
$$
$$
\frac{8}{8}x^2 + \frac{25}{8}x^2 = \frac{33}{8}x^2
$$
- Combine:
$$
-\frac{67}{15}x^4 + \frac{33}{8}x^2
$$
- Answer: $ -\frac{67}{15}x^4 + \frac{33}{8}x^2 $

21. $ \left(-\frac{5}{3}x^4 + \frac{5}{4}x^3\right) + \left(\frac{1}{4}x^4 - 2x^3\right) $
- Combine like terms:
$$
-\frac{5}{3}x^4 + \frac{1}{4}x^4 + \frac{5}{4}x^3 - 2x^3
$$
- Simplify the $x^4$ terms. Find the LCD of 3 and 4, which is 12:
$$
-\frac{5}{3}x^4 = -\frac{20}{12}x^4, \quad \frac{1}{4}x^4 = \frac{3}{12}x^4
$$
$$
-\frac{20}{12}x^4 + \frac{3}{12}x^4 = -\frac{17}{12}x^4
$$
- Simplify the $x^3$ terms. Find the LCD of 4 and 1, which is 4:
$$
\frac{5}{4}x^3 = \frac{5}{4}x^3, \quad -2x^3 = -\frac{8}{4}x^3
$$
$$
\frac{5}{4}x^3 - \frac{8}{4}x^3 = -\frac{3}{4}x^3
$$
- Combine:
$$
-\frac{17}{12}x^4 - \frac{3}{4}x^3
$$
- Answer: $ -\frac{17}{12}x^4 - \frac{3}{4}x^3 $

22. $ \left(-\frac{10}{7}x^4 - \frac{4}{5}\right) - \left(\frac{29}{6}x^4 - \frac{1}{5}x^4\right) $
- Distribute the negative sign:
$$
-\frac{10}{7}x^4 - \frac{4}{5} - \frac{29}{6}x^4 + \frac{1}{5}x^4
$$
- Combine like terms:
$$
-\frac{10}{7}x^4 - \frac{29}{6}x^4 + \frac{1}{5}x^4 - \frac{4}{5}
$$
- Simplify the $x^4$ terms. Find the LCD of 7, 6, and 5, which is 210:
$$
-\frac{10}{7}x^4 = -\frac{300}{210}x^4, \quad -\frac{29}{6}x^4 = -\frac{1015}{210}x^4, \quad \frac{1}{5}x^4 = \frac{42}{210}x^4
$$
$$
-\frac{300}{210}x^4 - \frac{1015}{210}x^4 + \frac{42}{210}x^4 = \frac{-300 - 1015 + 42}{210}x^4 = \frac{-1273}{210}x^4
$$
- Simplify the constant terms. Find the LCD of 5, which is 5:
$$
-\frac{4}{5} = -\frac{4}{5}, \quad 0 = 0
$$
$$
-\frac{4}{5}
$$
- Combine:
$$
-\frac{1273}{210}x^4 - \frac{4}{5}
$$
- Answer: $ -\frac{1273}{210}x^4 - \frac{4}{5} $

Final Answers:

1. $ -\frac{17}{30} $
2. $ \frac{1}{6} $
3. $ -\frac{41}{12} $
4. $ \frac{1}{8} $
5. $ \frac{7}{24} $
6. $ -\frac{35}{8} $
7. $ \frac{73}{14} $
8. $ \frac{3}{2} $
9. $ -\frac{7}{2} $
10. $ \frac{9}{4} $
11. $ \frac{1}{8} $
12. $ -\frac{29}{12} $
13. $ \frac{257}{84} $
14. $ -\frac{313}{24} $
15. $ \frac{593}{210} $
16. $ \frac{41}{21} $
17. $ a^2 + \frac{1}{3} $
18. $ \frac{77}{12}n + \frac{5}{6} $
19. $ \frac{14}{3}x^4 + \frac{31}{6} $
20. $ -\frac{67}{15}x^4 + \frac{33}{8}x^2 $
21. $ -\frac{17}{12}x^4 - \frac{3}{4}x^3 $
22. $ -\frac{1273}{210}x^4 - \frac{4}{5} $

$$
\boxed{\text{See detailed solutions above.}}
$$