Algebra Examples

$\frac{1}{2} (x - 1) - (x - 3) = \frac{1}{3} (x + 3) + \frac{1}{6}$

Step 1

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

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Apply the distributive property.

$\frac{1}{2} x + \frac{1}{2} \cdot - 1 - (x - 3) = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.1.2

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} + \frac{1}{2} \cdot - 1 - (x - 3) = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.1.3

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{x}{2} + \frac{- 1}{2} - (x - 3) = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.1.4

Move the negative in front of the fraction.

$\frac{x}{2} - \frac{1}{2} - (x - 3) = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.1.5

Apply the distributive property.

$\frac{x}{2} - \frac{1}{2} - x - - 3 = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.1.6

Multiply $- 1$ by $- 3$ .

$\frac{x}{2} - \frac{1}{2} - x + 3 = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.2

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

$\frac{x}{2} - x \cdot \frac{2}{2} - \frac{1}{2} + 3 = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.3

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

$\frac{x}{2} + \frac{- x \cdot 2}{2} - \frac{1}{2} + 3 = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{x - x \cdot 2}{2} - \frac{1}{2} + 3 = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.5

Combine the numerators over the common denominator.

$3 + \frac{x - x \cdot 2 - 1}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.6

Multiply $2$ by $- 1$ .

$3 + \frac{x - 2 x - 1}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.7

Subtract $2 x$ from $x$ .

$3 + \frac{- x - 1}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.8

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

$3 \cdot \frac{2}{2} + \frac{- x - 1}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.9

Simplify terms.

Tap for more steps...

Step 1.9.1

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

$\frac{3 \cdot 2}{2} + \frac{- x - 1}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.9.2

Combine the numerators over the common denominator.

$\frac{3 \cdot 2 - x - 1}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.10

Simplify the numerator.

Tap for more steps...

Step 1.10.1

Multiply $3$ by $2$ .

$\frac{6 - x - 1}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.10.2

Subtract $1$ from $6$ .

$\frac{- x + 5}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.11

Simplify with factoring out.

Tap for more steps...

Step 1.11.1

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

$\frac{- (x) + 5}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.11.2

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{- (x) - 1 (- 5)}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.11.3

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$\frac{- (x - 5)}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.11.4

Simplify the expression.

Tap for more steps...

Step 1.11.4.1

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{- 1 (x - 5)}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 1.11.4.2

Move the negative in front of the fraction.

$- \frac{x - 5}{2} = \frac{1}{3} \cdot (x + 3) + \frac{1}{6}$

Step 2

Simplify $\frac{1}{3} \cdot (x + 3) + \frac{1}{6}$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Apply the distributive property.

$- \frac{x - 5}{2} = \frac{1}{3} x + \frac{1}{3} \cdot 3 + \frac{1}{6}$

Step 2.1.2

Combine $\frac{1}{3}$ and $x$ .

$- \frac{x - 5}{2} = \frac{x}{3} + \frac{1}{3} \cdot 3 + \frac{1}{6}$

Step 2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.1.3.1

Cancel the common factor.

$- \frac{x - 5}{2} = \frac{x}{3} + \frac{1}{3} \cdot 3 + \frac{1}{6}$

Step 2.1.3.2

Rewrite the expression.

$- \frac{x - 5}{2} = \frac{x}{3} + 1 + \frac{1}{6}$

Step 2.2

Simplify the expression.

Tap for more steps...

Step 2.2.1

Write $1$ as a fraction with a common denominator.

$- \frac{x - 5}{2} = \frac{x}{3} + \frac{6}{6} + \frac{1}{6}$

Step 2.2.2

Combine the numerators over the common denominator.

$- \frac{x - 5}{2} = \frac{x}{3} + \frac{6 + 1}{6}$

Step 2.2.3

Add $6$ and $1$ .

$- \frac{x - 5}{2} = \frac{x}{3} + \frac{7}{6}$

Step 3

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 3.1

Subtract $\frac{x}{3}$ from both sides of the equation.

$- \frac{x - 5}{2} - \frac{x}{3} = \frac{7}{6}$

Step 3.2

To write $- \frac{x - 5}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{x - 5}{2} \cdot \frac{3}{3} - \frac{x}{3} = \frac{7}{6}$

Step 3.3

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

$- \frac{x - 5}{2} \cdot \frac{3}{3} - \frac{x}{3} \cdot \frac{2}{2} = \frac{7}{6}$

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.4.1

Multiply $\frac{x - 5}{2}$ by $\frac{3}{3}$ .

$- \frac{(x - 5) \cdot 3}{2 \cdot 3} - \frac{x}{3} \cdot \frac{2}{2} = \frac{7}{6}$

Step 3.4.2

Multiply $2$ by $3$ .

$- \frac{(x - 5) \cdot 3}{6} - \frac{x}{3} \cdot \frac{2}{2} = \frac{7}{6}$

Step 3.4.3

Multiply $\frac{x}{3}$ by $\frac{2}{2}$ .

$- \frac{(x - 5) \cdot 3}{6} - \frac{x \cdot 2}{3 \cdot 2} = \frac{7}{6}$

Step 3.4.4

Multiply $3$ by $2$ .

$- \frac{(x - 5) \cdot 3}{6} - \frac{x \cdot 2}{6} = \frac{7}{6}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- (x - 5) \cdot 3 - x \cdot 2}{6} = \frac{7}{6}$

Step 3.6

Simplify the numerator.

Tap for more steps...

Step 3.6.1

Apply the distributive property.

$\frac{(- x - - 5) \cdot 3 - x \cdot 2}{6} = \frac{7}{6}$

Step 3.6.2

Multiply $- 1$ by $- 5$ .

$\frac{(- x + 5) \cdot 3 - x \cdot 2}{6} = \frac{7}{6}$

Step 3.6.3

Apply the distributive property.

$\frac{- x \cdot 3 + 5 \cdot 3 - x \cdot 2}{6} = \frac{7}{6}$

Step 3.6.4

Multiply $3$ by $- 1$ .

$\frac{- 3 x + 5 \cdot 3 - x \cdot 2}{6} = \frac{7}{6}$

Step 3.6.5

Multiply $5$ by $3$ .

$\frac{- 3 x + 15 - x \cdot 2}{6} = \frac{7}{6}$

Step 3.6.6

Multiply $2$ by $- 1$ .

$\frac{- 3 x + 15 - 2 x}{6} = \frac{7}{6}$

Step 3.6.7

Subtract $2 x$ from $- 3 x$ .

$\frac{- 5 x + 15}{6} = \frac{7}{6}$

Step 3.6.8

Factor $5$ out of $- 5 x + 15$ .

Tap for more steps...

Step 3.6.8.1

Factor $5$ out of $- 5 x$ .

$\frac{5 (- x) + 15}{6} = \frac{7}{6}$

Step 3.6.8.2

Factor $5$ out of $15$ .

$\frac{5 (- x) + 5 (3)}{6} = \frac{7}{6}$

Step 3.6.8.3

Factor $5$ out of $5 (- x) + 5 (3)$ .

$\frac{5 (- x + 3)}{6} = \frac{7}{6}$

Step 3.7

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

$\frac{5 (- (x) + 3)}{6} = \frac{7}{6}$

Step 3.8

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

$\frac{5 (- (x) - 1 (- 3))}{6} = \frac{7}{6}$

Step 3.9

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

$\frac{5 (- (x - 3))}{6} = \frac{7}{6}$

Step 3.10

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

$\frac{5 (- 1 (x - 3))}{6} = \frac{7}{6}$

Step 3.11

Move the negative in front of the fraction.

$- \frac{5 (x - 3)}{6} = \frac{7}{6}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- 5 (x - 3) = 7$

Step 5

Divide each term in $- 5 (x - 3) = 7$ by $- 5$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $- 5 (x - 3) = 7$ by $- 5$ .

$\frac{- 5 (x - 3)}{- 5} = \frac{7}{- 5}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $- 5$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$\frac{- 5 (x - 3)}{- 5} = \frac{7}{- 5}$

Step 5.2.1.2

Divide $x - 3$ by $1$ .

$x - 3 = \frac{7}{- 5}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Move the negative in front of the fraction.

$x - 3 = - \frac{7}{5}$

Step 6

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

Tap for more steps...

Step 6.1

Add $3$ to both sides of the equation.

$x = - \frac{7}{5} + 3$

Step 6.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = - \frac{7}{5} + 3 \cdot \frac{5}{5}$

Step 6.3

Combine $3$ and $\frac{5}{5}$ .

$x = - \frac{7}{5} + \frac{3 \cdot 5}{5}$

Step 6.4

Combine the numerators over the common denominator.

$x = \frac{- 7 + 3 \cdot 5}{5}$

Step 6.5

Simplify the numerator.

Tap for more steps...

Step 6.5.1

Multiply $3$ by $5$ .

$x = \frac{- 7 + 15}{5}$

Step 6.5.2

Add $- 7$ and $15$ .

$x = \frac{8}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{5}$

Decimal Form:

$x = 1.6$

Mixed Number Form:

$x = 1 \frac{3}{5}$