To express the given fractions as a single fraction, we will follow these steps:
1. Find the least common multiple (LCM) of the denominators.
2. Convert each fraction to have this common denominator.
3. Combine the fractions by adding them up.
4. Simplify the result if possible.
Let's start with the given fractions:
[tex]\[
\frac{x-1}{2} + \frac{x+2}{4} + \frac{x}{5}
\][/tex]
### Step 1: Find the least common multiple (LCM) of the denominators
The denominators are [tex]\(2\)[/tex], [tex]\(4\)[/tex], and [tex]\(5\)[/tex].
- The prime factorization of [tex]\(2\)[/tex] is [tex]\(2\)[/tex].
- The prime factorization of [tex]\(4\)[/tex] is [tex]\(2^2\)[/tex].
- The prime factorization of [tex]\(5\)[/tex] is [tex]\(5\)[/tex].
The LCM of [tex]\(2\)[/tex], [tex]\(4\)[/tex], and [tex]\(5\)[/tex] is the smallest number that is a multiple of all of these numbers. This can be found by taking the highest power of each prime that appears in the factorizations:
[tex]\[
\text{LCM}(2, 4, 5) = 2^2 \cdot 5 = 4 \cdot 5 = 20
\][/tex]
So, the LCM of [tex]\(2\)[/tex], [tex]\(4\)[/tex], and [tex]\(5\)[/tex] is [tex]\(20\)[/tex].
### Step 2: Convert each fraction to have the common denominator of 20
We'll convert each fraction:
- For [tex]\(\frac{x-1}{2}\)[/tex]:
[tex]\[
\frac{x-1}{2} = \frac{(x-1) \cdot 10}{2 \cdot 10} = \frac{10(x-1)}{20} = \frac{10x-10}{20}
\][/tex]
- For [tex]\(\frac{x+2}{4}\)[/tex]:
[tex]\[
\frac{x+2}{4} = \frac{(x+2) \cdot 5}{4 \cdot 5} = \frac{5(x+2)}{20} = \frac{5x+10}{20}
\][/tex]
- For [tex]\(\frac{x}{5}\)[/tex]:
[tex]\[
\frac{x}{5} = \frac{x \cdot 4}{5 \cdot 4} = \frac{4x}{20}
\][/tex]
### Step 3: Add the fractions
Now that all fractions have a common denominator of [tex]\(20\)[/tex], we can combine them:
[tex]\[
\frac{10x - 10}{20} + \frac{5x + 10}{20} + \frac{4x}{20}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{(10x - 10) + (5x + 10) + 4x}{20}
\][/tex]
Combine like terms in the numerator:
[tex]\[
10x - 10 + 5x + 10 + 4x = (10x + 5x + 4x) + (-10 + 10) = 19x + 0 = 19x
\][/tex]
So, the combined fraction is:
[tex]\[
\frac{19x}{20}
\][/tex]
### Step 4: Simplify the fraction if possible
In this case, the fraction [tex]\(\frac{19x}{20}\)[/tex] is already in its simplest form.
### Final Answer
The given expression [tex]\(\frac{x-1}{2} + \frac{x+2}{4} + \frac{x}{5}\)[/tex] as a single fraction is:
[tex]\[
\frac{19x}{20}
\][/tex]
