Algebra Examples

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{1}{4} x - (\frac{1}{2} x + \frac{1}{3}) - \frac{5}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1

Simplify $\frac{5}{8} + \frac{3}{2} \cdot (\frac{1}{4} x - (\frac{1}{2} x + \frac{1}{3}) - \frac{5}{4})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Combine $\frac{1}{4}$ and $x$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{x}{4} - (\frac{1}{2} x + \frac{1}{3}) - \frac{5}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.1.2

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

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{x}{4} - (\frac{x}{2} + \frac{1}{3}) - \frac{5}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.1.3

Apply the distributive property.

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{x}{4} - \frac{x}{2} - \frac{1}{3} - \frac{5}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.2

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

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{x}{4} - \frac{x}{2} \cdot \frac{2}{2} - \frac{1}{3} - \frac{5}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.3.1

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

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{x}{4} - \frac{x \cdot 2}{2 \cdot 2} - \frac{1}{3} - \frac{5}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.3.2

Multiply $2$ by $2$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{x}{4} - \frac{x \cdot 2}{4} - \frac{1}{3} - \frac{5}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.4

Combine the numerators over the common denominator.

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{x - x \cdot 2}{4} - \frac{1}{3} - \frac{5}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{x - x \cdot 2 - 5}{4} + \frac{- 1}{3}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.6

Multiply $2$ by $- 1$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{x - 2 x - 5}{4} + \frac{- 1}{3}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.7

Subtract $2 x$ from $x$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{- x - 5}{4} + \frac{- 1}{3}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.8

Move the negative in front of the fraction.

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{- x - 5}{4} - \frac{1}{3}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.9

To write $\frac{- x - 5}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{- x - 5}{4} \cdot \frac{3}{3} - \frac{1}{3}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.10

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{- x - 5}{4} \cdot \frac{3}{3} - \frac{1}{3} \cdot \frac{4}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.11

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

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Step 1.1.11.1

Multiply $\frac{- x - 5}{4}$ by $\frac{3}{3}$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{(- x - 5) \cdot 3}{4 \cdot 3} - \frac{1}{3} \cdot \frac{4}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.11.2

Multiply $4$ by $3$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{(- x - 5) \cdot 3}{12} - \frac{1}{3} \cdot \frac{4}{4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.11.3

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{(- x - 5) \cdot 3}{12} - \frac{4}{3 \cdot 4}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.11.4

Multiply $3$ by $4$ .

$\frac{5}{8} + \frac{3}{2} \cdot (\frac{(- x - 5) \cdot 3}{12} - \frac{4}{12}) = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.12

Combine the numerators over the common denominator.

$\frac{5}{8} + \frac{3}{2} \cdot \frac{(- x - 5) \cdot 3 - 4}{12} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.13

Simplify the numerator.

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Step 1.1.13.1

Apply the distributive property.

$\frac{5}{8} + \frac{3}{2} \cdot \frac{- x \cdot 3 - 5 \cdot 3 - 4}{12} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.13.2

Multiply $3$ by $- 1$ .

$\frac{5}{8} + \frac{3}{2} \cdot \frac{- 3 x - 5 \cdot 3 - 4}{12} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.13.3

Multiply $- 5$ by $3$ .

$\frac{5}{8} + \frac{3}{2} \cdot \frac{- 3 x - 15 - 4}{12} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.13.4

Subtract $4$ from $- 15$ .

$\frac{5}{8} + \frac{3}{2} \cdot \frac{- 3 x - 19}{12} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.14

Cancel the common factor of $3$ .

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Step 1.1.14.1

Factor $3$ out of $12$ .

$\frac{5}{8} + \frac{3}{2} \cdot \frac{- 3 x - 19}{3 (4)} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.14.2

Cancel the common factor.

$\frac{5}{8} + \frac{3}{2} \cdot \frac{- 3 x - 19}{3 \cdot 4} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.14.3

Rewrite the expression.

$\frac{5}{8} + \frac{1}{2} \cdot \frac{- 3 x - 19}{4} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.15

Multiply $\frac{1}{2}$ by $\frac{- 3 x - 19}{4}$ .

$\frac{5}{8} + \frac{- 3 x - 19}{2 \cdot 4} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.1.16

Multiply $2$ by $4$ .

$\frac{5}{8} + \frac{- 3 x - 19}{8} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.2

Simplify terms.

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Step 1.2.1

Combine the numerators over the common denominator.

$\frac{5 - 3 x - 19}{8} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.2.2

Subtract $19$ from $5$ .

$\frac{- 3 x - 14}{8} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.2.3

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

$\frac{- (3 x) - 14}{8} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.2.4

Rewrite $- 14$ as $- 1 (14)$ .

$\frac{- (3 x) - 1 (14)}{8} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.2.5

Factor $- 1$ out of $- (3 x) - 1 (14)$ .

$\frac{- (3 x + 14)}{8} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Rewrite $- (3 x + 14)$ as $- 1 (3 x + 14)$ .

$\frac{- 1 (3 x + 14)}{8} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 1.2.6.2

Move the negative in front of the fraction.

$- \frac{3 x + 14}{8} = \frac{3}{4} \cdot (x - \frac{1}{6}) - x$

Step 2

Simplify $\frac{3}{4} \cdot (x - \frac{1}{6}) - x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- \frac{3 x + 14}{8} = \frac{3}{4} x + \frac{3}{4} (- \frac{1}{6}) - x$

Step 2.1.2

Combine $\frac{3}{4}$ and $x$ .

$- \frac{3 x + 14}{8} = \frac{3 x}{4} + \frac{3}{4} (- \frac{1}{6}) - x$

Step 2.1.3

Cancel the common factor of $3$ .

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Step 2.1.3.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$- \frac{3 x + 14}{8} = \frac{3 x}{4} + \frac{3}{4} \cdot \frac{- 1}{6} - x$

Step 2.1.3.2

Factor $3$ out of $6$ .

$- \frac{3 x + 14}{8} = \frac{3 x}{4} + \frac{3}{4} \cdot \frac{- 1}{3 (2)} - x$

Step 2.1.3.3

Cancel the common factor.

$- \frac{3 x + 14}{8} = \frac{3 x}{4} + \frac{3}{4} \cdot \frac{- 1}{3 \cdot 2} - x$

Step 2.1.3.4

Rewrite the expression.

$- \frac{3 x + 14}{8} = \frac{3 x}{4} + \frac{1}{4} \cdot \frac{- 1}{2} - x$

Step 2.1.4

Multiply $\frac{1}{4}$ by $\frac{- 1}{2}$ .

$- \frac{3 x + 14}{8} = \frac{3 x}{4} + \frac{- 1}{4 \cdot 2} - x$

Step 2.1.5

Multiply $4$ by $2$ .

$- \frac{3 x + 14}{8} = \frac{3 x}{4} + \frac{- 1}{8} - x$

Step 2.1.6

Move the negative in front of the fraction.

$- \frac{3 x + 14}{8} = \frac{3 x}{4} - \frac{1}{8} - x$

Step 2.2

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

$- \frac{3 x + 14}{8} = \frac{3 x}{4} - x \cdot \frac{4}{4} - \frac{1}{8}$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $- x$ and $\frac{4}{4}$ .

$- \frac{3 x + 14}{8} = \frac{3 x}{4} + \frac{- x \cdot 4}{4} - \frac{1}{8}$

Step 2.3.2

Combine the numerators over the common denominator.

$- \frac{3 x + 14}{8} = \frac{3 x - x \cdot 4}{4} - \frac{1}{8}$

Step 2.4

Simplify each term.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Factor $x$ out of $3 x - x \cdot 4$ .

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Step 2.4.1.1.1

Factor $x$ out of $3 x$ .

$- \frac{3 x + 14}{8} = \frac{x \cdot 3 - x \cdot 4}{4} - \frac{1}{8}$

Step 2.4.1.1.2

Factor $x$ out of $- x \cdot 4$ .

$- \frac{3 x + 14}{8} = \frac{x \cdot 3 + x (- 1 \cdot 4)}{4} - \frac{1}{8}$

Step 2.4.1.1.3

Factor $x$ out of $x \cdot 3 + x (- 1 \cdot 4)$ .

$- \frac{3 x + 14}{8} = \frac{x (3 - 1 \cdot 4)}{4} - \frac{1}{8}$

Step 2.4.1.2

Multiply $- 1$ by $4$ .

$- \frac{3 x + 14}{8} = \frac{x (3 - 4)}{4} - \frac{1}{8}$

Step 2.4.1.3

Subtract $4$ from $3$ .

$- \frac{3 x + 14}{8} = \frac{x \cdot - 1}{4} - \frac{1}{8}$

Step 2.4.2

Move $- 1$ to the left of $x$ .

$- \frac{3 x + 14}{8} = \frac{- 1 \cdot x}{4} - \frac{1}{8}$

Step 2.4.3

Move the negative in front of the fraction.

$- \frac{3 x + 14}{8} = - \frac{x}{4} - \frac{1}{8}$

Step 3

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

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Step 3.1

Add $\frac{x}{4}$ to both sides of the equation.

$- \frac{3 x + 14}{8} + \frac{x}{4} = - \frac{1}{8}$

Step 3.2

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

$- \frac{3 x + 14}{8} + \frac{x}{4} \cdot \frac{2}{2} = - \frac{1}{8}$

Step 3.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

$- \frac{3 x + 14}{8} + \frac{x \cdot 2}{4 \cdot 2} = - \frac{1}{8}$

Step 3.3.2

Multiply $4$ by $2$ .

$- \frac{3 x + 14}{8} + \frac{x \cdot 2}{8} = - \frac{1}{8}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- (3 x + 14) + x \cdot 2}{8} = - \frac{1}{8}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Apply the distributive property.

$\frac{- (3 x) - 1 \cdot 14 + x \cdot 2}{8} = - \frac{1}{8}$

Step 3.5.2

Multiply $3$ by $- 1$ .

$\frac{- 3 x - 1 \cdot 14 + x \cdot 2}{8} = - \frac{1}{8}$

Step 3.5.3

Multiply $- 1$ by $14$ .

$\frac{- 3 x - 14 + x \cdot 2}{8} = - \frac{1}{8}$

Step 3.5.4

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

$\frac{- 3 x - 14 + 2 \cdot x}{8} = - \frac{1}{8}$

Step 3.5.5

Add $- 3 x$ and $2 x$ .

$\frac{- x - 14}{8} = - \frac{1}{8}$

Step 3.6

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

$\frac{- (x) - 14}{8} = - \frac{1}{8}$

Step 3.7

Rewrite $- 14$ as $- 1 (14)$ .

$\frac{- (x) - 1 (14)}{8} = - \frac{1}{8}$

Step 3.8

Factor $- 1$ out of $- (x) - 1 (14)$ .

$\frac{- (x + 14)}{8} = - \frac{1}{8}$

Step 3.9

Rewrite $- (x + 14)$ as $- 1 (x + 14)$ .

$\frac{- 1 (x + 14)}{8} = - \frac{1}{8}$

Step 3.10

Move the negative in front of the fraction.

$- \frac{x + 14}{8} = - \frac{1}{8}$

Step 4

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

$- (x + 14) = - 1$

Step 5

Simplify $- (x + 14)$ .

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Step 5.1

Apply the distributive property.

$- x - 1 \cdot 14 = - 1$

Step 5.2

Multiply $- 1$ by $14$ .

$- x - 14 = - 1$

Step 6

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

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Step 6.1

Add $14$ to both sides of the equation.

$- x = - 1 + 14$

Step 6.2

Add $- 1$ and $14$ .

$- x = 13$

Step 7

Divide each term in $- x = 13$ by $- 1$ and simplify.

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Step 7.1

Divide each term in $- x = 13$ by $- 1$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Dividing two negative values results in a positive value.

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

Step 7.2.2

Divide $x$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $13$ by $- 1$ .

$x = - 13$