Algebra Examples

$\frac{x + 2}{4} + \frac{x - 4}{2} - \frac{x - 3}{3} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1

Simplify $\frac{x + 2}{4} + \frac{x - 4}{2} - \frac{x - 3}{3}$ .

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Step 1.1

Find the common denominator.

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Step 1.1.1

Multiply $\frac{x + 2}{4}$ by $\frac{3}{3}$ .

$\frac{x + 2}{4} \cdot \frac{3}{3} + \frac{x - 4}{2} - \frac{x - 3}{3} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.1.2

Multiply $\frac{x + 2}{4}$ by $\frac{3}{3}$ .

$\frac{(x + 2) \cdot 3}{4 \cdot 3} + \frac{x - 4}{2} - \frac{x - 3}{3} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.1.3

Multiply $\frac{x - 4}{2}$ by $\frac{6}{6}$ .

$\frac{(x + 2) \cdot 3}{4 \cdot 3} + \frac{x - 4}{2} \cdot \frac{6}{6} - \frac{x - 3}{3} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.1.4

Multiply $\frac{x - 4}{2}$ by $\frac{6}{6}$ .

$\frac{(x + 2) \cdot 3}{4 \cdot 3} + \frac{(x - 4) \cdot 6}{2 \cdot 6} - \frac{x - 3}{3} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.1.5

Multiply $\frac{x - 3}{3}$ by $\frac{4}{4}$ .

$\frac{(x + 2) \cdot 3}{4 \cdot 3} + \frac{(x - 4) \cdot 6}{2 \cdot 6} - (\frac{x - 3}{3} \cdot \frac{4}{4}) = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.1.6

Multiply $\frac{x - 3}{3}$ by $\frac{4}{4}$ .

$\frac{(x + 2) \cdot 3}{4 \cdot 3} + \frac{(x - 4) \cdot 6}{2 \cdot 6} - \frac{(x - 3) \cdot 4}{3 \cdot 4} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.1.7

Reorder the factors of $4 \cdot 3$ .

$\frac{(x + 2) \cdot 3}{3 \cdot 4} + \frac{(x - 4) \cdot 6}{2 \cdot 6} - \frac{(x - 3) \cdot 4}{3 \cdot 4} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.1.8

Multiply $3$ by $4$ .

$\frac{(x + 2) \cdot 3}{12} + \frac{(x - 4) \cdot 6}{2 \cdot 6} - \frac{(x - 3) \cdot 4}{3 \cdot 4} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.1.9

Multiply $2$ by $6$ .

$\frac{(x + 2) \cdot 3}{12} + \frac{(x - 4) \cdot 6}{12} - \frac{(x - 3) \cdot 4}{3 \cdot 4} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.1.10

Multiply $3$ by $4$ .

$\frac{(x + 2) \cdot 3}{12} + \frac{(x - 4) \cdot 6}{12} - \frac{(x - 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{(x + 2) \cdot 3 + (x - 4) \cdot 6 - (x - 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the distributive property.

$\frac{x \cdot 3 + 2 \cdot 3 + (x - 4) \cdot 6 - (x - 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.2

Move $3$ to the left of $x$ .

$\frac{3 \cdot x + 2 \cdot 3 + (x - 4) \cdot 6 - (x - 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.3

Multiply $2$ by $3$ .

$\frac{3 \cdot x + 6 + (x - 4) \cdot 6 - (x - 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.4

Apply the distributive property.

$\frac{3 x + 6 + x \cdot 6 - 4 \cdot 6 - (x - 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.5

Move $6$ to the left of $x$ .

$\frac{3 x + 6 + 6 \cdot x - 4 \cdot 6 - (x - 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.6

Multiply $- 4$ by $6$ .

$\frac{3 x + 6 + 6 \cdot x - 24 - (x - 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.7

Apply the distributive property.

$\frac{3 x + 6 + 6 x - 24 + (- x - - 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.8

Multiply $- 1$ by $- 3$ .

$\frac{3 x + 6 + 6 x - 24 + (- x + 3) \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.9

Apply the distributive property.

$\frac{3 x + 6 + 6 x - 24 - x \cdot 4 + 3 \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.10

Multiply $4$ by $- 1$ .

$\frac{3 x + 6 + 6 x - 24 - 4 x + 3 \cdot 4}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.3.11

Multiply $3$ by $4$ .

$\frac{3 x + 6 + 6 x - 24 - 4 x + 12}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.4

Simplify by adding terms.

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Step 1.4.1

Add $3 x$ and $6 x$ .

$\frac{9 x + 6 - 24 - 4 x + 12}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.4.2

Subtract $4 x$ from $9 x$ .

$\frac{5 x + 6 - 24 + 12}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.4.3

Simplify by adding and subtracting.

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Step 1.4.3.1

Subtract $24$ from $6$ .

$\frac{5 x - 18 + 12}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 1.4.3.2

Add $- 18$ and $12$ .

$\frac{5 x - 6}{12} = \frac{x + 1}{12} - \frac{x - 1}{2}$

Step 2

Simplify $\frac{x + 1}{12} - \frac{x - 1}{2}$ .

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Step 2.1

To write $- \frac{x - 1}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5 x - 6}{12} = \frac{x + 1}{12} - \frac{x - 1}{2} \cdot \frac{6}{6}$

Step 2.2

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1

Multiply $\frac{x - 1}{2}$ by $\frac{6}{6}$ .

$\frac{5 x - 6}{12} = \frac{x + 1}{12} - \frac{(x - 1) \cdot 6}{2 \cdot 6}$

Step 2.2.2

Multiply $2$ by $6$ .

$\frac{5 x - 6}{12} = \frac{x + 1}{12} - \frac{(x - 1) \cdot 6}{12}$

Step 2.3

Combine the numerators over the common denominator.

$\frac{5 x - 6}{12} = \frac{x + 1 - (x - 1) \cdot 6}{12}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{5 x - 6}{12} = \frac{x + 1 + (- x - - 1) \cdot 6}{12}$

Step 2.4.2

Multiply $- 1$ by $- 1$ .

$\frac{5 x - 6}{12} = \frac{x + 1 + (- x + 1) \cdot 6}{12}$

Step 2.4.3

Apply the distributive property.

$\frac{5 x - 6}{12} = \frac{x + 1 - x \cdot 6 + 1 \cdot 6}{12}$

Step 2.4.4

Multiply $6$ by $- 1$ .

$\frac{5 x - 6}{12} = \frac{x + 1 - 6 x + 1 \cdot 6}{12}$

Step 2.4.5

Multiply $6$ by $1$ .

$\frac{5 x - 6}{12} = \frac{x + 1 - 6 x + 6}{12}$

Step 2.4.6

Subtract $6 x$ from $x$ .

$\frac{5 x - 6}{12} = \frac{- 5 x + 1 + 6}{12}$

Step 2.4.7

Add $1$ and $6$ .

$\frac{5 x - 6}{12} = \frac{- 5 x + 7}{12}$

Step 2.5

Simplify with factoring out.

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Step 2.5.1

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

$\frac{5 x - 6}{12} = \frac{- (5 x) + 7}{12}$

Step 2.5.2

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{5 x - 6}{12} = \frac{- (5 x) - 1 (- 7)}{12}$

Step 2.5.3

Factor $- 1$ out of $- (5 x) - 1 (- 7)$ .

$\frac{5 x - 6}{12} = \frac{- (5 x - 7)}{12}$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Rewrite $- (5 x - 7)$ as $- 1 (5 x - 7)$ .

$\frac{5 x - 6}{12} = \frac{- 1 (5 x - 7)}{12}$

Step 2.5.4.2

Move the negative in front of the fraction.

$\frac{5 x - 6}{12} = - \frac{5 x - 7}{12}$

Step 3

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

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Step 3.1

Add $\frac{5 x - 7}{12}$ to both sides of the equation.

$\frac{5 x - 6}{12} + \frac{5 x - 7}{12} = 0$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 x - 6 + 5 x - 7}{12} = 0$

Step 3.3

Add $5 x$ and $5 x$ .

$\frac{10 x - 6 - 7}{12} = 0$

Step 3.4

Subtract $7$ from $- 6$ .

$\frac{10 x - 13}{12} = 0$

Step 4

Set the numerator equal to zero.

$10 x - 13 = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Add $13$ to both sides of the equation.

$10 x = 13$

Step 5.2

Divide each term in $10 x = 13$ by $10$ and simplify.

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Step 5.2.1

Divide each term in $10 x = 13$ by $10$ .

$\frac{10 x}{10} = \frac{13}{10}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $10$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{13}{10}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{10}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13}{10}$

Decimal Form:

$x = 1.3$

Mixed Number Form:

$x = 1 \frac{3}{10}$