Question

1. \\(\frac{7x}{15} - \frac{4x}{15}\\)
2. \\(\frac{2}{n} - \frac{6}{n}\\)
3. \\(\frac{2x}{3} - \frac{x - 1}{3}\\)
4. \\(\frac{3a}{10} - \frac{5a + 1}{10}\\)
5. \\(\frac{x + 3}{4} - \frac{9x + 11}{4}\\)
6. \\(\frac{2}{x + 3} + \frac{5}{x + 3}\\)
7. \\(\frac{x}{x - 2} + \frac{1}{x - 2}\\)
8. \\(\frac{n}{n - 2} - \frac{3n - 1}{n - 2}\\)
9. \\(\frac{5}{4y - 1} - \frac{3}{1 - 4y}\\)
10. \\(\frac{5c}{c - d} + \frac{5d}{d - c}\\)
11. \\(\frac{2}{x} + \frac{3}{x^2}\\)
12. \\(\frac{8}{ab} - \frac{6}{ab^2}\\)
13. \\(\frac{4}{3xy} + \frac{3}{2x^2}\\)
14. \\(\frac{4}{5(x + 1)} + \frac{x}{x + 1}\\)
15. \\(\frac{1 + 2x}{4} + \frac{1 + 2x}{6}\\)
16. \\(\frac{5b + 2}{8} + \frac{3b - 4}{16}\\)
17. \\(\frac{a - 6}{15} + \frac{4 - a}{10}\\)
18. \\(\frac{5n - 2}{12} - \frac{3(n - 3)}{8}\\)
19. \\(\frac{4x + 3}{3} - \frac{7x}{4} + \frac{x - 3}{6}\\)
20. \\(\frac{2m}{5} - \frac{m + 4}{4} + \frac{2 - m}{10}\\)

Explanation:

Let's solve problem 2: \(\frac{7x}{15}-\frac{4x}{15}\)

Step 1: Subtract the numerators

Since the denominators are the same, we subtract the numerators: \(\frac{7x - 4x}{15}\)

Step 2: Simplify the numerator

Simplify \(7x - 4x\) to get \(3x\), so we have \(\frac{3x}{15}\)

Step 3: Simplify the fraction

Divide numerator and denominator by 3: \(\frac{x}{5}\)

Now problem 3: \(\frac{2}{n}-\frac{6}{n}\)

Step 1: Subtract the numerators

Same denominator, so \(\frac{2 - 6}{n}\)

Step 2: Simplify the numerator

\(2 - 6 = -4\), so \(\frac{-4}{n}\) or \(-\frac{4}{n}\)

Problem 5: \(\frac{2x}{3}-\frac{x - 1}{3}\)

Step 1: Subtract the numerators

\(\frac{2x-(x - 1)}{3}\)

Step 2: Simplify the numerator

Distribute the negative sign: \(2x - x + 1 = x + 1\), so \(\frac{x + 1}{3}\)

Problem 6: \(\frac{3a}{10}-\frac{5a + 1}{10}\)

Step 1: Subtract the numerators

\(\frac{3a-(5a + 1)}{10}\)

Step 2: Simplify the numerator

\(3a - 5a - 1 = -2a - 1\), so \(\frac{-2a - 1}{10}\) or \(-\frac{2a + 1}{10}\)

Problem 8: \(\frac{x + 3}{4}-\frac{9x + 11}{4}\)

Step 1: Subtract the numerators

\(\frac{(x + 3)-(9x + 11)}{4}\)

Step 2: Simplify the numerator

\(x + 3 - 9x - 11 = -8x - 8\), factor out -8: \(-8(x + 1)\), so \(\frac{-8(x + 1)}{4} = -2(x + 1) = -2x - 2\)

Problem 9: \(\frac{2}{x + 3}+\frac{5}{x + 3}\)

Step 1: Add the numerators

Same denominator, so \(\frac{2 + 5}{x + 3}\)

Step 2: Simplify the numerator

\(2 + 5 = 7\), so \(\frac{7}{x + 3}\)

Problem 11: \(\frac{x}{x - 2}+\frac{1}{x - 2}\)

Step 1: Add the numerators

\(\frac{x + 1}{x - 2}\)

Problem 12: \(\frac{n}{n - 2}-\frac{3n - 1}{n - 2}\)

Step 1: Subtract the numerators

\(\frac{n-(3n - 1)}{n - 2}\)

Step 2: Simplify the numerator

\(n - 3n + 1 = -2n + 1\), so \(\frac{-2n + 1}{n - 2}\) or \(\frac{1 - 2n}{n - 2}\)

Problem 14: \(\frac{5}{4y - 1}-\frac{3}{1 - 4y}\)

Step 1: Rewrite the second denominator

Notice that \(1 - 4y = -(4y - 1)\), so \(\frac{3}{1 - 4y} = -\frac{3}{4y - 1}\)

Step 2: Subtract

\(\frac{5}{4y - 1}-(-\frac{3}{4y - 1}) = \frac{5 + 3}{4y - 1} = \frac{8}{4y - 1}\)

Problem 15: \(\frac{5c}{c - d}+\frac{5d}{d - c}\)

Step 1: Rewrite the second denominator

\(d - c = -(c - d)\), so \(\frac{5d}{d - c} = -\frac{5d}{c - d}\)

Step 2: Add

\(\frac{5c}{c - d}-\frac{5d}{c - d} = \frac{5c - 5d}{c - d}\)

Step 3: Factor the numerator

Factor out 5: \(\frac{5(c - d)}{c - d} = 5\) (assuming \(c eq d\))

Problem 17: \(\frac{2}{x}+\frac{3}{x^2}\)

Step 1: Find a common denominator

The common denominator is \(x^2\), so rewrite \(\frac{2}{x}\) as \(\frac{2x}{x^2}\)

Step 2: Add the fractions

\(\frac{2x}{x^2}+\frac{3}{x^2} = \frac{2x + 3}{x^2}\)

Problem 18: \(\frac{8}{ab}-\frac{6}{ab^2}\)

Step 1: Find a common denominator

Common denominator is \(ab^2\), rewrite \(\frac{8}{ab}\) as \(\frac{8b}{ab^2}\)

Step 2: Subtract the fractions

\(\frac{8b}{ab^2}-\frac{6}{ab^2} = \frac{8b - 6}{ab^2}\)

Step 3: Factor the numerator

Factor out 2: \(\frac{2(4b - 3)}{ab^2}\)

Problem 20: \(\frac{4}{3xy}+\frac{3}{2x^2}\)

Step 1: Find a common denominator

Common denominator is \(6x^2y\), rewrite \(\frac{4}{3xy}\) as \(\frac{8x}{6x^2y}\) and \(\frac{3}{2x^2}\) as \(\frac{9y}{6x^2y}\)

Step 2: Add the fractions

\(\frac{8x + 9y}{6x^2y}\)

Problem 21: \(\frac{4}{5(x + 1)}+\frac{x}{x + 1}\)

Step 1: Find a common denominator

Common denominator is \(5(x + 1)\), rewrite \(\frac{x}{x + 1}\) as \(\frac{5x}{5(x + 1)}\)

Step 2: Add the fractions

\(\frac{4 + 5x}{5(x + 1)}\) or \(\frac{5x + 4}{5(x + 1)}\)

Problem 23: \(\frac{1 + 2x}{4}+\frac{1 + 2x}{6}\)

Step 1: Find a common denominator

Common denominator is 12, rewrite \(\frac{1 + 2x}{4}\) as \(\frac{3(1 + 2x)}{12}\) and \(\frac{1 + 2x}{6}\) as \(\frac{2(1 + 2x)}{12}\)

Step 2: Add the fractions

\(\frac{3(1 + 2x) + 2(1 + 2x)}{12} = \frac{3 + 6x + 2 + 4x}{12} = \frac{5 + 10x}{12}\)

Step 3: Factor the numerator

Factor out 5: \(\frac{5(1 + 2x)}{12}\)

Problem 24: \(\frac{5b + 2}{8}+\frac{3b - 4}{16}\)

Step 1: Find a common denominator

Common denominator is 16, rewrite \(\frac{5b + 2}{8}\) as \(\frac{2(5b + 2)}{16}\)

Step 2: Add the fractions

\(\frac{2(5b + 2) + 3b - 4}{16} = \frac{10b + 4 + 3b - 4}{16} = \frac{13b}{16}\)

Problem 26: \(\frac{a - 6}{15}+\frac{4 - a}{10}\)

Step 1: Find a common denominator

Common denominator is 30, rewrite \(\frac{a - 6}{15}\) as \(\frac{2(a - 6)}{30}\) and \(\frac{4 - a}{10}\) as \(\frac{3(4 - a)}{30}\)

Step 2: Add the fractions

\(\frac{2(a - 6) + 3(4 - a)}{30} = \frac{2a - 12 + 12 - 3a}{30} = \frac{-a}{30}\) or \(-\frac{a}{30}\)

Problem 27: \(\frac{5n - 2}{12}-\frac{3(n - 3)}{8}\)

Step 1: Find a common denominator

Common denominator is 24, rewrite \(\frac{5n - 2}{12}\) as \(\frac{2(5n - 2)}{24}\) and \(\frac{3(n - 3)}{8}\) as \(\frac{9(n - 3)}{24}\)

Step 2: Subtract the fractions

\(\frac{2(5n - 2)-9(n - 3)}{24} = \frac{10n - 4 - 9n + 27}{24} = \frac{n + 23}{24}\)

Problem 29: \(\frac{4x + 3}{3}-\frac{7x}{4}+\frac{x - 3}{6}\)

Step 1: Find a common denominator

Common denominator is 12, rewrite each fraction:
\(\frac{4x + 3}{3} = \frac{4(4x + 3)}{12}\), \(\frac{7x}{4} = \frac{21x}{12}\), \(\frac{x - 3}{6} = \frac{2(x - 3)}{12}\)

Step 2: Combine the fractions

\(\frac{4(4x + 3)-21x + 2(x - 3)}{12} = \frac{16x + 12 - 21x + 2x - 6}{12} = \frac{-3x + 6}{12}\)

Step 3: Simplify the fraction

Factor out -3: \(\frac{-3(x - 2)}{12} = \frac{-(x - 2)}{4} = \frac{-x + 2}{4}\) or \(\frac{2 - x}{4}\)

Problem 30: \(\frac{2m}{5}-\frac{m + 4}{4}+\frac{2 - m}{10}\)

Step 1: Find a common denominator

Common denominator is 20, rewrite each fraction:
\(\frac{2m}{5} = \frac{8m}{20}\), \(\frac{m + 4}{4} = \frac{5(m + 4)}{20}\), \(\frac{2 - m}{10} = \frac{2(2 - m)}{20}\)

Step 2: Combine the fractions

\(\frac{8m - 5(m + 4) + 2(2 - m)}{20} = \frac{8m - 5m - 20 + 4 - 2m}{20} = \frac{m - 16}{20}\)

Answers:

2. \(\boldsymbol{\frac{x}{5}}\)
3. \(\boldsymbol{-\frac{4}{n}}\)
5. \(\boldsymbol{\frac{x + 1}{3}}\)
6. \(\boldsymbol{-\frac{2a + 1}{10}}\)
8. \(\boldsymbol{-2x - 2}\)
9. \(\boldsymbol{\frac{7}{x + 3}}\)
11. \(\boldsymbol{\frac{x + 1}{x - 2}}\)
12. \(\boldsymbol{\frac{1 - 2n}{n - 2}}\)
14. \(\boldsymbol{\frac{8}{4y - 1}}\)
15. \(\boldsymbol{5}\) (for \(c eq d\))
17. \(\boldsymbol{\frac{2x + 3}{x^2}}\)
18. \(\boldsymbol{\frac{2(4b - 3)}{ab^2}}\)
20. \(\boldsymbol{\frac{8x + 9y}{6x^2y}}\)
21. \(\boldsymbol{\frac{5x + 4}{5(x + 1)}}\)
23. \(\boldsymbol{\frac{5(1 + 2x)}{12}}\)
24. \(\boldsymbol{\frac{13b}{16}}\)
26. \(\boldsymbol{-\frac{a}{30}}\)
27. \(\boldsymbol{\frac{n + 23}{24}}\)
29. \(\boldsymbol{\frac{2 - x}{4}}\)
30. \(\boldsymbol{\frac{m - 16}{20}}\)

Answer:

s:

1. \(\boldsymbol{\frac{x}{5}}\)
2. \(\boldsymbol{-\frac{4}{n}}\)
3. \(\boldsymbol{\frac{x + 1}{3}}\)
4. \(\boldsymbol{-\frac{2a + 1}{10}}\)
5. \(\boldsymbol{-2x - 2}\)
6. \(\boldsymbol{\frac{7}{x + 3}}\)
7. \(\boldsymbol{\frac{x + 1}{x - 2}}\)
8. \(\boldsymbol{\frac{1 - 2n}{n - 2}}\)
9. \(\boldsymbol{\frac{8}{4y - 1}}\)
10. \(\boldsymbol{5}\) (for \(c

eq d\))

1. \(\boldsymbol{\frac{2x + 3}{x^2}}\)
2. \(\boldsymbol{\frac{2(4b - 3)}{ab^2}}\)
3. \(\boldsymbol{\frac{8x + 9y}{6x^2y}}\)
4. \(\boldsymbol{\frac{5x + 4}{5(x + 1)}}\)
5. \(\boldsymbol{\frac{5(1 + 2x)}{12}}\)
6. \(\boldsymbol{\frac{13b}{16}}\)
7. \(\boldsymbol{-\frac{a}{30}}\)
8. \(\boldsymbol{\frac{n + 23}{24}}\)
9. \(\boldsymbol{\frac{2 - x}{4}}\)
10. \(\boldsymbol{\frac{m - 16}{20}}\)