Precalculus Examples

$\frac{3}{40} \cdot (x - 1) - \frac{1}{8} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1

Simplify $\frac{3}{40} \cdot (x - 1) - \frac{1}{8}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{3}{40} x + \frac{3}{40} \cdot - 1 - \frac{1}{8} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.1.2

Combine $\frac{3}{40}$ and $x$ .

$\frac{3 x}{40} + \frac{3}{40} \cdot - 1 - \frac{1}{8} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.1.3

Multiply $\frac{3}{40} \cdot - 1$ .

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Step 1.1.3.1

Combine $\frac{3}{40}$ and $- 1$ .

$\frac{3 x}{40} + \frac{3 \cdot - 1}{40} - \frac{1}{8} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.1.3.2

Multiply $3$ by $- 1$ .

$\frac{3 x}{40} + \frac{- 3}{40} - \frac{1}{8} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.1.4

Move the negative in front of the fraction.

$\frac{3 x}{40} - \frac{3}{40} - \frac{1}{8} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.2

To write $- \frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3 x}{40} - \frac{3}{40} - \frac{1}{8} \cdot \frac{5}{5} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.3

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{8}$ by $\frac{5}{5}$ .

$\frac{3 x}{40} - \frac{3}{40} - \frac{5}{8 \cdot 5} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.3.2

Multiply $8$ by $5$ .

$\frac{3 x}{40} - \frac{3}{40} - \frac{5}{40} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.4

Simplify the expression.

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Step 1.4.1

Combine the numerators over the common denominator.

$\frac{3 x}{40} + \frac{- 3 - 5}{40} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.4.2

Subtract $5$ from $- 3$ .

$\frac{3 x}{40} + \frac{- 8}{40} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.5

Simplify each term.

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Step 1.5.1

Cancel the common factor of $- 8$ and $40$ .

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Step 1.5.1.1

Factor $8$ out of $- 8$ .

$\frac{3 x}{40} + \frac{8 (- 1)}{40} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.5.1.2

Cancel the common factors.

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Step 1.5.1.2.1

Factor $8$ out of $40$ .

$\frac{3 x}{40} + \frac{8 \cdot - 1}{8 \cdot 5} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.5.1.2.2

Cancel the common factor.

$\frac{3 x}{40} + \frac{8 \cdot - 1}{8 \cdot 5} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.5.1.2.3

Rewrite the expression.

$\frac{3 x}{40} + \frac{- 1}{5} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 1.5.2

Move the negative in front of the fraction.

$\frac{3 x}{40} - \frac{1}{5} = \frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 2

Simplify $\frac{6}{10} - \frac{1}{20} \cdot (2 x + 2)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Cancel the common factor of $6$ and $10$ .

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Step 2.1.1.1

Factor $2$ out of $6$ .

$\frac{3 x}{40} - \frac{1}{5} = \frac{2 (3)}{10} - \frac{1}{20} \cdot (2 x + 2)$

Step 2.1.1.2

Cancel the common factors.

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Step 2.1.1.2.1

Factor $2$ out of $10$ .

$\frac{3 x}{40} - \frac{1}{5} = \frac{2 \cdot 3}{2 \cdot 5} - \frac{1}{20} \cdot (2 x + 2)$

Step 2.1.1.2.2

Cancel the common factor.

$\frac{3 x}{40} - \frac{1}{5} = \frac{2 \cdot 3}{2 \cdot 5} - \frac{1}{20} \cdot (2 x + 2)$

Step 2.1.1.2.3

Rewrite the expression.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} - \frac{1}{20} \cdot (2 x + 2)$

Step 2.1.2

Apply the distributive property.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} - \frac{1}{20} (2 x) - \frac{1}{20} \cdot 2$

Step 2.1.3

Cancel the common factor of $2$ .

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Step 2.1.3.1

Move the leading negative in $- \frac{1}{20}$ into the numerator.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- 1}{20} (2 x) - \frac{1}{20} \cdot 2$

Step 2.1.3.2

Factor $2$ out of $20$ .

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- 1}{2 (10)} (2 x) - \frac{1}{20} \cdot 2$

Step 2.1.3.3

Factor $2$ out of $2 x$ .

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- 1}{2 (10)} (2 (x)) - \frac{1}{20} \cdot 2$

Step 2.1.3.4

Cancel the common factor.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- 1}{2 \cdot 10} (2 x) - \frac{1}{20} \cdot 2$

Step 2.1.3.5

Rewrite the expression.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- 1}{10} x - \frac{1}{20} \cdot 2$

Step 2.1.4

Combine $\frac{- 1}{10}$ and $x$ .

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- x}{10} - \frac{1}{20} \cdot 2$

Step 2.1.5

Cancel the common factor of $2$ .

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Step 2.1.5.1

Move the leading negative in $- \frac{1}{20}$ into the numerator.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- x}{10} + \frac{- 1}{20} \cdot 2$

Step 2.1.5.2

Factor $2$ out of $20$ .

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- x}{10} + \frac{- 1}{2 (10)} \cdot 2$

Step 2.1.5.3

Cancel the common factor.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- x}{10} + \frac{- 1}{2 \cdot 10} \cdot 2$

Step 2.1.5.4

Rewrite the expression.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} + \frac{- x}{10} + \frac{- 1}{10}$

Step 2.1.6

Simplify each term.

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Step 2.1.6.1

Move the negative in front of the fraction.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} - \frac{x}{10} + \frac{- 1}{10}$

Step 2.1.6.2

Move the negative in front of the fraction.

$\frac{3 x}{40} - \frac{1}{5} = \frac{3}{5} - \frac{x}{10} - \frac{1}{10}$

Step 2.2

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

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{3}{5} \cdot \frac{2}{2} - \frac{1}{10}$

Step 2.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{3 \cdot 2}{5 \cdot 2} - \frac{1}{10}$

Step 2.3.2

Multiply $5$ by $2$ .

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{3 \cdot 2}{10} - \frac{1}{10}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{3 \cdot 2 - 1}{10}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $3$ by $2$ .

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{6 - 1}{10}$

Step 2.5.2

Subtract $1$ from $6$ .

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{5}{10}$

Step 2.6

Cancel the common factor of $5$ and $10$ .

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Step 2.6.1

Factor $5$ out of $5$ .

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{5 (1)}{10}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $5$ out of $10$ .

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{5 \cdot 1}{5 \cdot 2}$

Step 2.6.2.2

Cancel the common factor.

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{5 \cdot 1}{5 \cdot 2}$

Step 2.6.2.3

Rewrite the expression.

$\frac{3 x}{40} - \frac{1}{5} = - \frac{x}{10} + \frac{1}{2}$

Step 3

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

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Step 3.1

Add $\frac{x}{10}$ to both sides of the equation.

$\frac{3 x}{40} - \frac{1}{5} + \frac{x}{10} = \frac{1}{2}$

Step 3.2

To write $\frac{x}{10}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 x}{40} + \frac{x}{10} \cdot \frac{4}{4} - \frac{1}{5} = \frac{1}{2}$

Step 3.3

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{x}{10}$ by $\frac{4}{4}$ .

$\frac{3 x}{40} + \frac{x \cdot 4}{10 \cdot 4} - \frac{1}{5} = \frac{1}{2}$

Step 3.3.2

Multiply $10$ by $4$ .

$\frac{3 x}{40} + \frac{x \cdot 4}{40} - \frac{1}{5} = \frac{1}{2}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{3 x + x \cdot 4}{40} - \frac{1}{5} = \frac{1}{2}$

Step 3.5

Add $3 x$ and $x \cdot 4$ .

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Step 3.5.1

Reorder $x$ and $4$ .

$\frac{3 x + 4 \cdot x}{40} - \frac{1}{5} = \frac{1}{2}$

Step 3.5.2

Add $3 x$ and $4 \cdot x$ .

$\frac{7 x}{40} - \frac{1}{5} = \frac{1}{2}$

Step 4

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

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Step 4.1

Add $\frac{1}{5}$ to both sides of the equation.

$\frac{7 x}{40} = \frac{1}{2} + \frac{1}{5}$

Step 4.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{7 x}{40} = \frac{1}{2} \cdot \frac{5}{5} + \frac{1}{5}$

Step 4.3

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7 x}{40} = \frac{1}{2} \cdot \frac{5}{5} + \frac{1}{5} \cdot \frac{2}{2}$

Step 4.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$\frac{7 x}{40} = \frac{5}{2 \cdot 5} + \frac{1}{5} \cdot \frac{2}{2}$

Step 4.4.2

Multiply $2$ by $5$ .

$\frac{7 x}{40} = \frac{5}{10} + \frac{1}{5} \cdot \frac{2}{2}$

Step 4.4.3

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$\frac{7 x}{40} = \frac{5}{10} + \frac{2}{5 \cdot 2}$

Step 4.4.4

Multiply $5$ by $2$ .

$\frac{7 x}{40} = \frac{5}{10} + \frac{2}{10}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{7 x}{40} = \frac{5 + 2}{10}$

Step 4.6

Add $5$ and $2$ .

$\frac{7 x}{40} = \frac{7}{10}$

Step 5

Multiply both sides of the equation by $\frac{40}{7}$ .

$\frac{40}{7} \cdot \frac{7 x}{40} = \frac{40}{7} \cdot \frac{7}{10}$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{40}{7} \cdot \frac{7 x}{40}$ .

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Step 6.1.1.1

Cancel the common factor of $40$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{40}{7} \cdot \frac{7 x}{40} = \frac{40}{7} \cdot \frac{7}{10}$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 x) = \frac{40}{7} \cdot \frac{7}{10}$

Step 6.1.1.2

Cancel the common factor of $7$ .

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Step 6.1.1.2.1

Factor $7$ out of $7 x$ .

$\frac{1}{7} (7 (x)) = \frac{40}{7} \cdot \frac{7}{10}$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 x) = \frac{40}{7} \cdot \frac{7}{10}$

Step 6.1.1.2.3

Rewrite the expression.

$x = \frac{40}{7} \cdot \frac{7}{10}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{40}{7} \cdot \frac{7}{10}$ .

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Step 6.2.1.1

Cancel the common factor of $10$ .

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Step 6.2.1.1.1

Factor $10$ out of $40$ .

$x = \frac{10 (4)}{7} \cdot \frac{7}{10}$

Step 6.2.1.1.2

Cancel the common factor.

$x = \frac{10 \cdot 4}{7} \cdot \frac{7}{10}$

Step 6.2.1.1.3

Rewrite the expression.

$x = \frac{4}{7} \cdot 7$

Step 6.2.1.2

Cancel the common factor of $7$ .

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Step 6.2.1.2.1

Cancel the common factor.

$x = \frac{4}{7} \cdot 7$

Step 6.2.1.2.2

Rewrite the expression.

$x = 4$