Algebra Examples

$\frac{x + 4}{3} - \frac{x + 7}{2} = \frac{- 1}{6}$

Step 1

Simplify $\frac{x + 4}{3} - \frac{x + 7}{2}$ .

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Step 1.1

To write $\frac{x + 4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x + 4}{3} \cdot \frac{2}{2} - \frac{x + 7}{2} = \frac{- 1}{6}$

Step 1.2

To write $- \frac{x + 7}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x + 4}{3} \cdot \frac{2}{2} - \frac{x + 7}{2} \cdot \frac{3}{3} = \frac{- 1}{6}$

Step 1.3

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

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Step 1.3.1

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

$\frac{(x + 4) \cdot 2}{3 \cdot 2} - \frac{x + 7}{2} \cdot \frac{3}{3} = \frac{- 1}{6}$

Step 1.3.2

Multiply $3$ by $2$ .

$\frac{(x + 4) \cdot 2}{6} - \frac{x + 7}{2} \cdot \frac{3}{3} = \frac{- 1}{6}$

Step 1.3.3

Multiply $\frac{x + 7}{2}$ by $\frac{3}{3}$ .

$\frac{(x + 4) \cdot 2}{6} - \frac{(x + 7) \cdot 3}{2 \cdot 3} = \frac{- 1}{6}$

Step 1.3.4

Multiply $2$ by $3$ .

$\frac{(x + 4) \cdot 2}{6} - \frac{(x + 7) \cdot 3}{6} = \frac{- 1}{6}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{(x + 4) \cdot 2 - (x + 7) \cdot 3}{6} = \frac{- 1}{6}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{x \cdot 2 + 4 \cdot 2 - (x + 7) \cdot 3}{6} = \frac{- 1}{6}$

Step 1.5.2

Move $2$ to the left of $x$ .

$\frac{2 \cdot x + 4 \cdot 2 - (x + 7) \cdot 3}{6} = \frac{- 1}{6}$

Step 1.5.3

Multiply $4$ by $2$ .

$\frac{2 \cdot x + 8 - (x + 7) \cdot 3}{6} = \frac{- 1}{6}$

Step 1.5.4

Apply the distributive property.

$\frac{2 x + 8 + (- x - 1 \cdot 7) \cdot 3}{6} = \frac{- 1}{6}$

Step 1.5.5

Multiply $- 1$ by $7$ .

$\frac{2 x + 8 + (- x - 7) \cdot 3}{6} = \frac{- 1}{6}$

Step 1.5.6

Apply the distributive property.

$\frac{2 x + 8 - x \cdot 3 - 7 \cdot 3}{6} = \frac{- 1}{6}$

Step 1.5.7

Multiply $3$ by $- 1$ .

$\frac{2 x + 8 - 3 x - 7 \cdot 3}{6} = \frac{- 1}{6}$

Step 1.5.8

Multiply $- 7$ by $3$ .

$\frac{2 x + 8 - 3 x - 21}{6} = \frac{- 1}{6}$

Step 1.5.9

Subtract $3 x$ from $2 x$ .

$\frac{- x + 8 - 21}{6} = \frac{- 1}{6}$

Step 1.5.10

Subtract $21$ from $8$ .

$\frac{- x - 13}{6} = \frac{- 1}{6}$

Step 1.6

Simplify with factoring out.

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Step 1.6.1

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

$\frac{- (x) - 13}{6} = \frac{- 1}{6}$

Step 1.6.2

Rewrite $- 13$ as $- 1 (13)$ .

$\frac{- (x) - 1 (13)}{6} = \frac{- 1}{6}$

Step 1.6.3

Factor $- 1$ out of $- (x) - 1 (13)$ .

$\frac{- (x + 13)}{6} = \frac{- 1}{6}$

Step 1.6.4

Simplify the expression.

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Step 1.6.4.1

Rewrite $- (x + 13)$ as $- 1 (x + 13)$ .

$\frac{- 1 (x + 13)}{6} = \frac{- 1}{6}$

Step 1.6.4.2

Move the negative in front of the fraction.

$- \frac{x + 13}{6} = \frac{- 1}{6}$

Step 2

Move the negative in front of the fraction.

$- \frac{x + 13}{6} = - \frac{1}{6}$

Step 3

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

$- (x + 13) = - 1$

Step 4

Simplify $- (x + 13)$ .

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Step 4.1

Apply the distributive property.

$- x - 1 \cdot 13 = - 1$

Step 4.2

Multiply $- 1$ by $13$ .

$- x - 13 = - 1$

Step 5

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

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Step 5.1

Add $13$ to both sides of the equation.

$- x = - 1 + 13$

Step 5.2

Add $- 1$ and $13$ .

$- x = 12$

Step 6

Divide each term in $- x = 12$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- x = 12$ by $- 1$ .

$\frac{- x}{- 1} = \frac{12}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{12}{- 1}$

Step 6.2.2

Divide $x$ by $1$ .

$x = \frac{12}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $12$ by $- 1$ .

$x = - 12$