Pre-Algebra Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Multiply $\frac{1}{5} (3 x)$ .

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Step 1.1.2.1

Combine $3$ and $\frac{1}{5}$ .

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

Step 1.1.2.2

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

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

Step 1.1.3

Combine $\frac{1}{5}$ and $2$ .

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

Step 1.1.4

Apply the distributive property.

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

Step 1.1.5

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

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

Step 1.1.6

Multiply $- \frac{1}{3} \cdot 4$ .

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Step 1.1.6.1

Multiply $4$ by $- 1$ .

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

Step 1.1.6.2

Combine $- 4$ and $\frac{1}{3}$ .

$\frac{3 x}{5} + \frac{2}{5} - \frac{x}{3} + \frac{- 4}{3} = 6$

Step 1.1.7

Move the negative in front of the fraction.

$\frac{3 x}{5} + \frac{2}{5} - \frac{x}{3} - \frac{4}{3} = 6$

Step 1.2

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

$\frac{3 x}{5} \cdot \frac{3}{3} - \frac{x}{3} + \frac{2}{5} - \frac{4}{3} = 6$

Step 1.3

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

$\frac{3 x}{5} \cdot \frac{3}{3} - \frac{x}{3} \cdot \frac{5}{5} + \frac{2}{5} - \frac{4}{3} = 6$

Step 1.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

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

$\frac{3 x \cdot 3}{5 \cdot 3} - \frac{x}{3} \cdot \frac{5}{5} + \frac{2}{5} - \frac{4}{3} = 6$

Step 1.4.2

Multiply $5$ by $3$ .

$\frac{3 x \cdot 3}{15} - \frac{x}{3} \cdot \frac{5}{5} + \frac{2}{5} - \frac{4}{3} = 6$

Step 1.4.3

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

$\frac{3 x \cdot 3}{15} - \frac{x \cdot 5}{3 \cdot 5} + \frac{2}{5} - \frac{4}{3} = 6$

Step 1.4.4

Multiply $3$ by $5$ .

$\frac{3 x \cdot 3}{15} - \frac{x \cdot 5}{15} + \frac{2}{5} - \frac{4}{3} = 6$

Step 1.5

Combine the numerators over the common denominator.

$\frac{3 x \cdot 3 - x \cdot 5}{15} + \frac{2}{5} - \frac{4}{3} = 6$

Step 1.6

Find the common denominator.

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Step 1.6.1

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

$\frac{3 x \cdot 3 - x \cdot 5}{15} + \frac{2}{5} \cdot \frac{3}{3} - \frac{4}{3} = 6$

Step 1.6.2

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

$\frac{3 x \cdot 3 - x \cdot 5}{15} + \frac{2 \cdot 3}{5 \cdot 3} - \frac{4}{3} = 6$

Step 1.6.3

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

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

Step 1.6.4

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

$\frac{3 x \cdot 3 - x \cdot 5}{15} + \frac{2 \cdot 3}{5 \cdot 3} - \frac{4 \cdot 5}{3 \cdot 5} = 6$

Step 1.6.5

Reorder the factors of $5 \cdot 3$ .

$\frac{3 x \cdot 3 - x \cdot 5}{15} + \frac{2 \cdot 3}{3 \cdot 5} - \frac{4 \cdot 5}{3 \cdot 5} = 6$

Step 1.6.6

Multiply $3$ by $5$ .

$\frac{3 x \cdot 3 - x \cdot 5}{15} + \frac{2 \cdot 3}{15} - \frac{4 \cdot 5}{3 \cdot 5} = 6$

Step 1.6.7

Multiply $3$ by $5$ .

$\frac{3 x \cdot 3 - x \cdot 5}{15} + \frac{2 \cdot 3}{15} - \frac{4 \cdot 5}{15} = 6$

Step 1.7

Combine the numerators over the common denominator.

$\frac{3 x \cdot 3 - x \cdot 5 + 2 \cdot 3 - 4 \cdot 5}{15} = 6$

Step 1.8

Simplify each term.

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Step 1.8.1

Multiply $3$ by $3$ .

$\frac{9 x - x \cdot 5 + 2 \cdot 3 - 4 \cdot 5}{15} = 6$

Step 1.8.2

Multiply $5$ by $- 1$ .

$\frac{9 x - 5 x + 2 \cdot 3 - 4 \cdot 5}{15} = 6$

Step 1.8.3

Multiply $2$ by $3$ .

$\frac{9 x - 5 x + 6 - 4 \cdot 5}{15} = 6$

Step 1.8.4

Multiply $- 4$ by $5$ .

$\frac{9 x - 5 x + 6 - 20}{15} = 6$

Step 1.9

Simplify terms.

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Step 1.9.1

Subtract $5 x$ from $9 x$ .

$\frac{4 x + 6 - 20}{15} = 6$

Step 1.9.2

Subtract $20$ from $6$ .

$\frac{4 x - 14}{15} = 6$

Step 1.9.3

Factor $2$ out of $4 x - 14$ .

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Step 1.9.3.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) - 14}{15} = 6$

Step 1.9.3.2

Factor $2$ out of $- 14$ .

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

Step 1.9.3.3

Factor $2$ out of $2 (2 x) + 2 (- 7)$ .

$\frac{2 (2 x - 7)}{15} = 6$

Step 2

Multiply both sides by $15$ .

$\frac{2 (2 x - 7)}{15} \cdot 15 = 6 \cdot 15$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{2 (2 x - 7)}{15} \cdot 15$ .

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Step 3.1.1.1

Cancel the common factor of $15$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{2 (2 x - 7)}{15} \cdot 15 = 6 \cdot 15$

Step 3.1.1.1.2

Rewrite the expression.

$2 (2 x - 7) = 6 \cdot 15$

Step 3.1.1.2

Apply the distributive property.

$2 (2 x) + 2 \cdot - 7 = 6 \cdot 15$

Step 3.1.1.3

Multiply.

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Step 3.1.1.3.1

Multiply $2$ by $2$ .

$4 x + 2 \cdot - 7 = 6 \cdot 15$

Step 3.1.1.3.2

Multiply $2$ by $- 7$ .

$4 x - 14 = 6 \cdot 15$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $6$ by $15$ .

$4 x - 14 = 90$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Add $14$ to both sides of the equation.

$4 x = 90 + 14$

Step 4.1.2

Add $90$ and $14$ .

$4 x = 104$

Step 4.2

Divide each term in $4 x = 104$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 x = 104$ by $4$ .

$\frac{4 x}{4} = \frac{104}{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{104}{4}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{104}{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $104$ by $4$ .

$x = 26$