Solve for x (x-(x-1))/2=(1-(x-2))/3

$\frac{x - (x - 1)}{2} = \frac{1 - (x - 2)}{3}$

Step 1

Multiply both sides by $2$ .

$\frac{x - (x - 1)}{2} \cdot 2 = \frac{1 - (x - 2)}{3} \cdot 2$

Step 2

Simplify.

Simplify the left side.

Simplify $\frac{x - (x - 1)}{2} \cdot 2$ .

Simplify the numerator.

Apply the distributive property.

$\frac{x - x - - 1}{2} \cdot 2 = \frac{1 - (x - 2)}{3} \cdot 2$

Multiply $- 1$ by $- 1$ .

$\frac{x - x + 1}{2} \cdot 2 = \frac{1 - (x - 2)}{3} \cdot 2$

Subtract $x$ from $x$ .

$\frac{0 + 1}{2} \cdot 2 = \frac{1 - (x - 2)}{3} \cdot 2$

Add $0$ and $1$ .

$\frac{1}{2} \cdot 2 = \frac{1 - (x - 2)}{3} \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{2} \cdot 2 = \frac{1 - (x - 2)}{3} \cdot 2$

Rewrite the expression.

$1 = \frac{1 - (x - 2)}{3} \cdot 2$

Simplify the right side.

Simplify $\frac{1 - (x - 2)}{3} \cdot 2$ .

Simplify the numerator.

Apply the distributive property.

$1 = \frac{1 - x - - 2}{3} \cdot 2$

Multiply $- 1$ by $- 2$ .

$1 = \frac{1 - x + 2}{3} \cdot 2$

Add $1$ and $2$ .

$1 = \frac{- x + 3}{3} \cdot 2$

Simplify terms.

Combine $\frac{- x + 3}{3}$ and $2$ .

$1 = \frac{(- x + 3) \cdot 2}{3}$

Move $2$ to the left of $- x + 3$ .

$1 = \frac{2 \cdot (- x + 3)}{3}$

Factor $- 1$ out of $- x$ .

$1 = \frac{2 (- (x) + 3)}{3}$

Rewrite $3$ as $- 1 (- 3)$ .

$1 = \frac{2 (- (x) - 1 (- 3))}{3}$

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

$1 = \frac{2 (- (x - 3))}{3}$

Simplify the expression.

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$1 = \frac{2 (- 1 (x - 3))}{3}$

Move the negative in front of the fraction.

$1 = - \frac{2 (x - 3)}{3}$

Step 3

Solve for $x$ .

Rewrite the equation as $- \frac{2 (x - 3)}{3} = 1$ .

$- \frac{2 (x - 3)}{3} = 1$

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 (x - 3)}{3}) = - \frac{3}{2} \cdot 1$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{3}{2} (- \frac{2 (x - 3)}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 (x - 3)}{3}) = - \frac{3}{2} \cdot 1$

Move the leading negative in $- \frac{2 (x - 3)}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 (x - 3)}{3} = - \frac{3}{2} \cdot 1$

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 (x - 3)}{3} = - \frac{3}{2} \cdot 1$

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 (x - 3)}{3} = - \frac{3}{2} \cdot 1$

Rewrite the expression.

$\frac{- 1}{2} (- 2 (x - 3)) = - \frac{3}{2} \cdot 1$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2 (x - 3)$ .

$\frac{- 1}{2} (2 (- (x - 3))) = - \frac{3}{2} \cdot 1$

Cancel the common factor.

$\frac{- 1}{2} (2 (- (x - 3))) = - \frac{3}{2} \cdot 1$

Rewrite the expression.

$- - (x - 3) = - \frac{3}{2} \cdot 1$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (x - 3) = - \frac{3}{2} \cdot 1$

Multiply $x - 3$ by $1$ .

$x - 3 = - \frac{3}{2} \cdot 1$

Simplify the right side.

Multiply $- 1$ by $1$ .

$x - 3 = - \frac{3}{2}$

Move all terms not containing $x$ to the right side of the equation.

Add $3$ to both sides of the equation.

$x = - \frac{3}{2} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - \frac{3}{2} + 3 \cdot \frac{2}{2}$

Combine $3$ and $\frac{2}{2}$ .

$x = - \frac{3}{2} + \frac{3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$x = \frac{- 3 + 3 \cdot 2}{2}$

Simplify the numerator.

Multiply $3$ by $2$ .

$x = \frac{- 3 + 6}{2}$

Add $- 3$ and $6$ .

$x = \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{2}$

Decimal Form:

$x = 1.5$

Mixed Number Form:

$x = 1 \frac{1}{2}$

Solve for x (x-(x-1))/2=(1-(x-2))/3

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