Let's solve the expression step by step:
[tex]\[
\frac{x+1}{2} - \frac{x-2}{3}
\][/tex]
### Step 1: Identify the Least Common Denominator (LCD)
First, we need to find the least common denominator of the fractions. The denominators of the fractions are 2 and 3. The least common multiple (LCM) of 2 and 3 is 6. Thus, the LCD is 6.
### Step 2: Convert Each Fraction to Have the LCD as Denominator
We will convert each fraction to have the denominator of 6.
For the first fraction:
[tex]\[
\frac{x+1}{2} = \frac{x+1}{2} \times \frac{3}{3} = \frac{3(x+1)}{6} = \frac{3x + 3}{6}
\][/tex]
For the second fraction:
[tex]\[
\frac{x-2}{3} = \frac{x-2}{3} \times \frac{2}{2} = \frac{2(x-2)}{6} = \frac{2x - 4}{6}
\][/tex]
### Step 3: Subtract the Fractions with a Common Denominator
Now that both fractions have the same denominator, we can subtract them:
[tex]\[
\frac{3x + 3}{6} - \frac{2x - 4}{6}
\][/tex]
Since the denominators are the same, we can combine the numerators:
[tex]\[
\frac{(3x + 3) - (2x - 4)}{6}
\][/tex]
### Step 4: Simplify the Numerator
Simplify the expression in the numerator:
[tex]\[
(3x + 3) - (2x - 4) = 3x + 3 - 2x + 4 = 3x - 2x + 3 + 4 = x + 7
\][/tex]
### Step 5: Write the Final Answer
The simplified expression is:
[tex]\[
\frac{x + 7}{6}
\][/tex]
Hence, the solution to the given expression [tex]\(\frac{x+1}{2} - \frac{x-2}{3}\)[/tex] is:
[tex]\[
\frac{x + 7}{6}
\][/tex]
