Let's solve the equation step-by-step:
[tex]\[ x - 1 = \frac{x - 1}{2} - \frac{x - 2}{3} \][/tex]
1. Eliminate the fractions by finding a common denominator:
The least common multiple of 2 and 3 is 6. Multiply every term by 6 to clear the fractions.
[tex]\[ 6(x - 1) = 6 \left( \frac{x - 1}{2} \right) - 6 \left( \frac{x - 2}{3} \right) \][/tex]
2. Simplify each term:
[tex]\[ 6(x - 1) = 3(x - 1) - 2(x - 2) \][/tex]
3. Expand the terms:
[tex]\[ 6x - 6 = 3x - 3 - 2x + 4 \][/tex]
4. Combine like terms:
[tex]\[ 6x - 6 = 3x - 2x - 3 + 4 \][/tex]
Simplify the terms on the right hand side:
[tex]\[ 6x - 6 = x + 1 \][/tex]
5. Isolate x:
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ 6x - x - 6 = 1 \][/tex]
This simplifies to:
[tex]\[ 5x - 6 = 1 \][/tex]
6. Solve for [tex]\( x \)[/tex]:
Add 6 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 5x - 6 + 6 = 1 + 6 \][/tex]
[tex]\[ 5x = 7 \][/tex]
7. Divide both sides by 5:
[tex]\[ x = \frac{7}{5} \][/tex]
So the solution to the equation
[tex]\[ x - 1 = \frac{x - 1}{2} - \frac{x - 2}{3} \][/tex]
is
[tex]\[ x = \frac{7}{5}. \][/tex]
