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Let's solve each exercise step by step:
### (a) [tex]\(\frac{-1}{2} + \frac{3}{4}\)[/tex]
1. Find a common denominator for [tex]\(\frac{-1}{2}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex]. The common denominator is 4.
2. Convert [tex]\(\frac{-1}{2}\)[/tex] to [tex]\(\frac{-2}{4}\)[/tex].
3. Add [tex]\(\frac{-2}{4} + \frac{3}{4} = \frac{1}{4}\)[/tex].
Therefore, [tex]\(\frac{-1}{2} + \frac{3}{4} = \frac{1}{4}\)[/tex].
### (b) [tex]\(\frac{5}{9} - \frac{4}{3}\)[/tex]
1. Find a common denominator for [tex]\(\frac{5}{9}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex]. The common denominator is 9.
2. Convert [tex]\(\frac{4}{3}\)[/tex] to [tex]\(\frac{12}{9}\)[/tex].
3. Subtract [tex]\(\frac{5}{9} - \frac{12}{9} = \frac{-7}{9}\)[/tex].
Therefore, [tex]\(\frac{5}{9} - \frac{4}{3} = \frac{-7}{9}\)[/tex].
### (c) [tex]\(-1 \frac{1}{3} - 2 \frac{1}{4}\)[/tex]
1. Convert the mixed numbers to improper fractions:
- [tex]\(-1 \frac{1}{3} = \frac{-4}{3}\)[/tex]
- [tex]\(2 \frac{1}{4} = \frac{9}{4}\)[/tex]
2. Find a common denominator for [tex]\(\frac{-4}{3}\)[/tex] and [tex]\(\frac{9}{4}\)[/tex]. The common denominator is 12.
3. Convert the fractions:
- [tex]\(\frac{-4}{3} = \frac{-16}{12}\)[/tex]
- [tex]\(\frac{9}{4} = \frac{27}{12}\)[/tex]
4. Subtract [tex]\(\frac{-16}{12} - \frac{27}{12} = \frac{-43}{12}\)[/tex], which is approximately [tex]\(-3.5833\)[/tex].
Therefore, [tex]\(-1 \frac{1}{3} - 2 \frac{1}{4} = \frac{-43}{12}\)[/tex] or approximately [tex]\(-3.5833\)[/tex].
### (e) [tex]\(\frac{-3}{5} - \left(\frac{-7}{2}\right)\)[/tex]
1. Recognize that subtracting a negative is the same as adding a positive: [tex]\(\frac{-3}{5} + \frac{7}{2}\)[/tex].
2. Find a common denominator for [tex]\(\frac{-3}{5}\)[/tex] and [tex]\(\frac{7}{2}\)[/tex]. The common denominator is 10.
3. Convert the fractions:
- [tex]\(\frac{-3}{5} = \frac{-6}{10}\)[/tex]
- [tex]\(\frac{7}{2} = \frac{35}{10}\)[/tex]
4. Add [tex]\(\frac{-6}{10} + \frac{35}{10} = \frac{29}{10}\)[/tex], which is 2.9.
Therefore, [tex]\(\frac{-3}{5} - \left(\frac{-7}{2}\right) = 2.9\)[/tex].
### (f) [tex]\(16 - 2 \frac{3}{7}\)[/tex]
1. Convert the mixed number to an improper fraction:
- [tex]\(2 \frac{3}{7} = \frac{17}{7}\)[/tex]
2. Recognize that 16 is equivalent to [tex]\(\frac{112}{7}\)[/tex].
3. Subtract [tex]\(\frac{112}{7} - \frac{17}{7} = \frac{95}{7}\)[/tex], which is approximately 13.5714.
Therefore, [tex]\(16 - 2 \frac{3}{7} = \frac{95}{7}\)[/tex] or approximately 13.5714.
### (g) [tex]\(1 \frac{5}{7} - 4\)[/tex]
1. Convert the mixed number to an improper fraction:
- [tex]\(1 \frac{5}{7} = \frac{12}{7}\)[/tex]
2. Recognize that 4 is equivalent to [tex]\(\frac{28}{7}\)[/tex].
3. Subtract [tex]\(\frac{12}{7} - \frac{28}{7} = \frac{-16}{7}\)[/tex], which is approximately [tex]\(-2.2857\)[/tex].
Therefore, [tex]\(1 \frac{5}{7} - 4 = \frac{-16}{7}\)[/tex] or approximately [tex]\(-2.2857\)[/tex].
### (i) [tex]\(\frac{3}{19} + \frac{1}{2}\)[/tex]
1. Find a common denominator for [tex]\(\frac{3}{19}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]. The common denominator is 38.
2. Convert the fractions:
- [tex]\(\frac{3}{19} = \frac{6}{38}\)[/tex]
- [tex]\(\frac{1}{2} = \frac{19}{38}\)[/tex]
3. Add [tex]\(\frac{6}{38} + \frac{19}{38} = \frac{25}{38}\)[/tex], which is approximately 0.6579.
Therefore, [tex]\(\frac{3}{19} + \frac{1}{2} = \frac{25}{38}\)[/tex] or approximately 0.6579.
These are the solutions to the given exercises.
### (a) [tex]\(\frac{-1}{2} + \frac{3}{4}\)[/tex]
1. Find a common denominator for [tex]\(\frac{-1}{2}\)[/tex] and [tex]\(\frac{3}{4}\)[/tex]. The common denominator is 4.
2. Convert [tex]\(\frac{-1}{2}\)[/tex] to [tex]\(\frac{-2}{4}\)[/tex].
3. Add [tex]\(\frac{-2}{4} + \frac{3}{4} = \frac{1}{4}\)[/tex].
Therefore, [tex]\(\frac{-1}{2} + \frac{3}{4} = \frac{1}{4}\)[/tex].
### (b) [tex]\(\frac{5}{9} - \frac{4}{3}\)[/tex]
1. Find a common denominator for [tex]\(\frac{5}{9}\)[/tex] and [tex]\(\frac{4}{3}\)[/tex]. The common denominator is 9.
2. Convert [tex]\(\frac{4}{3}\)[/tex] to [tex]\(\frac{12}{9}\)[/tex].
3. Subtract [tex]\(\frac{5}{9} - \frac{12}{9} = \frac{-7}{9}\)[/tex].
Therefore, [tex]\(\frac{5}{9} - \frac{4}{3} = \frac{-7}{9}\)[/tex].
### (c) [tex]\(-1 \frac{1}{3} - 2 \frac{1}{4}\)[/tex]
1. Convert the mixed numbers to improper fractions:
- [tex]\(-1 \frac{1}{3} = \frac{-4}{3}\)[/tex]
- [tex]\(2 \frac{1}{4} = \frac{9}{4}\)[/tex]
2. Find a common denominator for [tex]\(\frac{-4}{3}\)[/tex] and [tex]\(\frac{9}{4}\)[/tex]. The common denominator is 12.
3. Convert the fractions:
- [tex]\(\frac{-4}{3} = \frac{-16}{12}\)[/tex]
- [tex]\(\frac{9}{4} = \frac{27}{12}\)[/tex]
4. Subtract [tex]\(\frac{-16}{12} - \frac{27}{12} = \frac{-43}{12}\)[/tex], which is approximately [tex]\(-3.5833\)[/tex].
Therefore, [tex]\(-1 \frac{1}{3} - 2 \frac{1}{4} = \frac{-43}{12}\)[/tex] or approximately [tex]\(-3.5833\)[/tex].
### (e) [tex]\(\frac{-3}{5} - \left(\frac{-7}{2}\right)\)[/tex]
1. Recognize that subtracting a negative is the same as adding a positive: [tex]\(\frac{-3}{5} + \frac{7}{2}\)[/tex].
2. Find a common denominator for [tex]\(\frac{-3}{5}\)[/tex] and [tex]\(\frac{7}{2}\)[/tex]. The common denominator is 10.
3. Convert the fractions:
- [tex]\(\frac{-3}{5} = \frac{-6}{10}\)[/tex]
- [tex]\(\frac{7}{2} = \frac{35}{10}\)[/tex]
4. Add [tex]\(\frac{-6}{10} + \frac{35}{10} = \frac{29}{10}\)[/tex], which is 2.9.
Therefore, [tex]\(\frac{-3}{5} - \left(\frac{-7}{2}\right) = 2.9\)[/tex].
### (f) [tex]\(16 - 2 \frac{3}{7}\)[/tex]
1. Convert the mixed number to an improper fraction:
- [tex]\(2 \frac{3}{7} = \frac{17}{7}\)[/tex]
2. Recognize that 16 is equivalent to [tex]\(\frac{112}{7}\)[/tex].
3. Subtract [tex]\(\frac{112}{7} - \frac{17}{7} = \frac{95}{7}\)[/tex], which is approximately 13.5714.
Therefore, [tex]\(16 - 2 \frac{3}{7} = \frac{95}{7}\)[/tex] or approximately 13.5714.
### (g) [tex]\(1 \frac{5}{7} - 4\)[/tex]
1. Convert the mixed number to an improper fraction:
- [tex]\(1 \frac{5}{7} = \frac{12}{7}\)[/tex]
2. Recognize that 4 is equivalent to [tex]\(\frac{28}{7}\)[/tex].
3. Subtract [tex]\(\frac{12}{7} - \frac{28}{7} = \frac{-16}{7}\)[/tex], which is approximately [tex]\(-2.2857\)[/tex].
Therefore, [tex]\(1 \frac{5}{7} - 4 = \frac{-16}{7}\)[/tex] or approximately [tex]\(-2.2857\)[/tex].
### (i) [tex]\(\frac{3}{19} + \frac{1}{2}\)[/tex]
1. Find a common denominator for [tex]\(\frac{3}{19}\)[/tex] and [tex]\(\frac{1}{2}\)[/tex]. The common denominator is 38.
2. Convert the fractions:
- [tex]\(\frac{3}{19} = \frac{6}{38}\)[/tex]
- [tex]\(\frac{1}{2} = \frac{19}{38}\)[/tex]
3. Add [tex]\(\frac{6}{38} + \frac{19}{38} = \frac{25}{38}\)[/tex], which is approximately 0.6579.
Therefore, [tex]\(\frac{3}{19} + \frac{1}{2} = \frac{25}{38}\)[/tex] or approximately 0.6579.
These are the solutions to the given exercises.
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