Answer :
To simplify the given fractions, we will follow a systematic approach step-by-step for each expression.
### 1) [tex]\(\frac{x-1}{2(x-4)} + \frac{x}{x-4}\)[/tex]
First, let's find a common denominator for the two fractions:
- The denominators are [tex]\(2(x-4)\)[/tex] and [tex]\(x-4\)[/tex].
- The least common denominator (LCD) is [tex]\(2(x-4)\)[/tex].
Rewriting both fractions with the common denominator:
[tex]\[ \frac{x-1}{2(x-4)} + \frac{2 \cdot x}{2(x-4)} \][/tex]
Combining the numerators over the common denominator:
[tex]\[ \frac{(x-1) + 2x}{2(x-4)} = \frac{3x - 1}{2(x-4)} \][/tex]
Therefore:
[tex]\[ \frac{x-1}{2(x-4)} + \frac{x}{x-4} = \frac{3x - 1}{2(x-4)} \][/tex]
### 2) [tex]\(\frac{7 p}{9(p+2 q)} - \frac{4 p}{3(p+2 q)}\)[/tex]
Both fractions already have a common denominator:
- The common denominator is [tex]\(9(p + 2q)\)[/tex].
Rewriting the second fraction to have the common denominator:
[tex]\[ \frac{7p}{9(p+2q)} - \frac{4p \cdot 3}{9(p+2q)} = \frac{7p}{9(p+2q)} - \frac{12p}{9(p+2q)} \][/tex]
Combining the numerators over the common denominator:
[tex]\[ \frac{7p - 12p}{9(p+2q)} = \frac{-5p}{9(p+2q)} \][/tex]
Therefore:
[tex]\[ \frac{7 p}{9(p+2 q)} - \frac{4 p}{3(p+2 q)} = \frac{-5p}{9(p+2q)} \][/tex]
### 3) [tex]\(\frac{a-2}{3a-12} - \frac{a-3}{2a-8}\)[/tex]
First, factor the denominators:
- [tex]\(3a - 12 = 3(a - 4)\)[/tex]
- [tex]\(2a - 8 = 2(a - 4)\)[/tex]
Now, let's rewrite each fraction:
[tex]\[ \frac{a-2}{3(a-4)} - \frac{a-3}{2(a-4)} \][/tex]
Find the least common denominator, which is [tex]\(6(a-4)\)[/tex]:
Rewrite each fraction with the common denominator:
[tex]\[ \frac{2(a-2)}{6(a-4)} - \frac{3(a-3)}{6(a-4)} \][/tex]
Combine the numerators over the common denominator:
[tex]\[ \frac{2(a-2) - 3(a-3)}{6(a-4)} \][/tex]
Simplify the numerator:
[tex]\[ = \frac{2a - 4 - 3a + 9}{6(a-4)} = \frac{-a + 5}{6(a-4)} = \frac{5 - a}{6(a-4)} \][/tex]
Therefore:
[tex]\[ \frac{a-2}{3a-12} - \frac{a-3}{2a-8} = \frac{5 - a}{6(a-4)} \][/tex]
### 4) [tex]\(\frac{3 m}{4 m-4} + \frac{2 m}{5-5 m}\)[/tex]
First, factor the denominators:
- [tex]\(4m - 4 = 4(m - 1)\)[/tex]
- Rewrite the second denominator so that [tex]\(5 - 5m = -5(m - 1)\)[/tex]
Now, let's rewrite each fraction:
[tex]\[ \frac{3m}{4(m-1)} + \frac{2m}{-5(m-1)} \][/tex]
Find the least common denominator, which is [tex]\(-20(m-1)\)[/tex]:
Rewrite each fraction with the common denominator:
[tex]\[ \frac{3m(-5)}{-20(m-1)} + \frac{2m \cdot 4}{-20(m-1)} \][/tex]
Combine the numerators over the common denominator:
[tex]\[ \frac{-15m + 8m}{-20(m-1)} = \frac{-7m}{-20(m-1)} = \frac{7m}{20(m-1)} \][/tex]
Therefore:
[tex]\[ \frac{3 m}{4 m-4} + \frac{2 m}{5-5 m} = \frac{7m}{20(m - 1)} \][/tex]
### Final simplified fractions:
1) [tex]\(\frac{3x - 1}{2(x-4)}\)[/tex]
2) [tex]\(\frac{-5p}{9(p+2q)}\)[/tex]
3) [tex]\(\frac{5 - a}{6(a-4)}\)[/tex]
4) [tex]\(\frac{7m}{20(m-1)}\)[/tex]
### 1) [tex]\(\frac{x-1}{2(x-4)} + \frac{x}{x-4}\)[/tex]
First, let's find a common denominator for the two fractions:
- The denominators are [tex]\(2(x-4)\)[/tex] and [tex]\(x-4\)[/tex].
- The least common denominator (LCD) is [tex]\(2(x-4)\)[/tex].
Rewriting both fractions with the common denominator:
[tex]\[ \frac{x-1}{2(x-4)} + \frac{2 \cdot x}{2(x-4)} \][/tex]
Combining the numerators over the common denominator:
[tex]\[ \frac{(x-1) + 2x}{2(x-4)} = \frac{3x - 1}{2(x-4)} \][/tex]
Therefore:
[tex]\[ \frac{x-1}{2(x-4)} + \frac{x}{x-4} = \frac{3x - 1}{2(x-4)} \][/tex]
### 2) [tex]\(\frac{7 p}{9(p+2 q)} - \frac{4 p}{3(p+2 q)}\)[/tex]
Both fractions already have a common denominator:
- The common denominator is [tex]\(9(p + 2q)\)[/tex].
Rewriting the second fraction to have the common denominator:
[tex]\[ \frac{7p}{9(p+2q)} - \frac{4p \cdot 3}{9(p+2q)} = \frac{7p}{9(p+2q)} - \frac{12p}{9(p+2q)} \][/tex]
Combining the numerators over the common denominator:
[tex]\[ \frac{7p - 12p}{9(p+2q)} = \frac{-5p}{9(p+2q)} \][/tex]
Therefore:
[tex]\[ \frac{7 p}{9(p+2 q)} - \frac{4 p}{3(p+2 q)} = \frac{-5p}{9(p+2q)} \][/tex]
### 3) [tex]\(\frac{a-2}{3a-12} - \frac{a-3}{2a-8}\)[/tex]
First, factor the denominators:
- [tex]\(3a - 12 = 3(a - 4)\)[/tex]
- [tex]\(2a - 8 = 2(a - 4)\)[/tex]
Now, let's rewrite each fraction:
[tex]\[ \frac{a-2}{3(a-4)} - \frac{a-3}{2(a-4)} \][/tex]
Find the least common denominator, which is [tex]\(6(a-4)\)[/tex]:
Rewrite each fraction with the common denominator:
[tex]\[ \frac{2(a-2)}{6(a-4)} - \frac{3(a-3)}{6(a-4)} \][/tex]
Combine the numerators over the common denominator:
[tex]\[ \frac{2(a-2) - 3(a-3)}{6(a-4)} \][/tex]
Simplify the numerator:
[tex]\[ = \frac{2a - 4 - 3a + 9}{6(a-4)} = \frac{-a + 5}{6(a-4)} = \frac{5 - a}{6(a-4)} \][/tex]
Therefore:
[tex]\[ \frac{a-2}{3a-12} - \frac{a-3}{2a-8} = \frac{5 - a}{6(a-4)} \][/tex]
### 4) [tex]\(\frac{3 m}{4 m-4} + \frac{2 m}{5-5 m}\)[/tex]
First, factor the denominators:
- [tex]\(4m - 4 = 4(m - 1)\)[/tex]
- Rewrite the second denominator so that [tex]\(5 - 5m = -5(m - 1)\)[/tex]
Now, let's rewrite each fraction:
[tex]\[ \frac{3m}{4(m-1)} + \frac{2m}{-5(m-1)} \][/tex]
Find the least common denominator, which is [tex]\(-20(m-1)\)[/tex]:
Rewrite each fraction with the common denominator:
[tex]\[ \frac{3m(-5)}{-20(m-1)} + \frac{2m \cdot 4}{-20(m-1)} \][/tex]
Combine the numerators over the common denominator:
[tex]\[ \frac{-15m + 8m}{-20(m-1)} = \frac{-7m}{-20(m-1)} = \frac{7m}{20(m-1)} \][/tex]
Therefore:
[tex]\[ \frac{3 m}{4 m-4} + \frac{2 m}{5-5 m} = \frac{7m}{20(m - 1)} \][/tex]
### Final simplified fractions:
1) [tex]\(\frac{3x - 1}{2(x-4)}\)[/tex]
2) [tex]\(\frac{-5p}{9(p+2q)}\)[/tex]
3) [tex]\(\frac{5 - a}{6(a-4)}\)[/tex]
4) [tex]\(\frac{7m}{20(m-1)}\)[/tex]
