Pre-Algebra Examples

$\frac{1}{2} \cdot (5 x + 4) - \frac{1}{3} \cdot (x + 7) = 4$

Step 1

Simplify $\frac{1}{2} \cdot (5 x + 4) - \frac{1}{3} \cdot (x + 7)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{1}{2} (5 x) + \frac{1}{2} \cdot 4 - \frac{1}{3} \cdot (x + 7) = 4$

Step 1.1.2

Multiply $\frac{1}{2} (5 x)$ .

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Step 1.1.2.1

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

$\frac{5}{2} x + \frac{1}{2} \cdot 4 - \frac{1}{3} \cdot (x + 7) = 4$

Step 1.1.2.2

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

$\frac{5 x}{2} + \frac{1}{2} \cdot 4 - \frac{1}{3} \cdot (x + 7) = 4$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Factor $2$ out of $4$ .

$\frac{5 x}{2} + \frac{1}{2} \cdot (2 (2)) - \frac{1}{3} \cdot (x + 7) = 4$

Step 1.1.3.2

Cancel the common factor.

$\frac{5 x}{2} + \frac{1}{2} \cdot (2 \cdot 2) - \frac{1}{3} \cdot (x + 7) = 4$

Step 1.1.3.3

Rewrite the expression.

$\frac{5 x}{2} + 2 - \frac{1}{3} \cdot (x + 7) = 4$

Step 1.1.4

Apply the distributive property.

$\frac{5 x}{2} + 2 - \frac{1}{3} x - \frac{1}{3} \cdot 7 = 4$

Step 1.1.5

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

$\frac{5 x}{2} + 2 - \frac{x}{3} - \frac{1}{3} \cdot 7 = 4$

Step 1.1.6

Multiply $- \frac{1}{3} \cdot 7$ .

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Step 1.1.6.1

Multiply $7$ by $- 1$ .

$\frac{5 x}{2} + 2 - \frac{x}{3} - 7 (\frac{1}{3}) = 4$

Step 1.1.6.2

Combine $- 7$ and $\frac{1}{3}$ .

$\frac{5 x}{2} + 2 - \frac{x}{3} + \frac{- 7}{3} = 4$

Step 1.1.7

Move the negative in front of the fraction.

$\frac{5 x}{2} + 2 - \frac{x}{3} - \frac{7}{3} = 4$

Step 1.2

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

$\frac{5 x}{2} \cdot \frac{3}{3} - \frac{x}{3} + 2 - \frac{7}{3} = 4$

Step 1.3

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

$\frac{5 x}{2} \cdot \frac{3}{3} - \frac{x}{3} \cdot \frac{2}{2} + 2 - \frac{7}{3} = 4$

Step 1.4

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

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Step 1.4.1

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

$\frac{5 x \cdot 3}{2 \cdot 3} - \frac{x}{3} \cdot \frac{2}{2} + 2 - \frac{7}{3} = 4$

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{5 x \cdot 3}{6} - \frac{x}{3} \cdot \frac{2}{2} + 2 - \frac{7}{3} = 4$

Step 1.4.3

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

$\frac{5 x \cdot 3}{6} - \frac{x \cdot 2}{3 \cdot 2} + 2 - \frac{7}{3} = 4$

Step 1.4.4

Multiply $3$ by $2$ .

$\frac{5 x \cdot 3}{6} - \frac{x \cdot 2}{6} + 2 - \frac{7}{3} = 4$

Step 1.5

Combine the numerators over the common denominator.

$\frac{5 x \cdot 3 - x \cdot 2}{6} + 2 - \frac{7}{3} = 4$

Step 1.6

Find the common denominator.

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Step 1.6.1

Write $2$ as a fraction with denominator $1$ .

$\frac{5 x \cdot 3 - x \cdot 2}{6} + \frac{2}{1} - \frac{7}{3} = 4$

Step 1.6.2

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

$\frac{5 x \cdot 3 - x \cdot 2}{6} + \frac{2}{1} \cdot \frac{6}{6} - \frac{7}{3} = 4$

Step 1.6.3

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

$\frac{5 x \cdot 3 - x \cdot 2}{6} + \frac{2 \cdot 6}{6} - \frac{7}{3} = 4$

Step 1.6.4

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

$\frac{5 x \cdot 3 - x \cdot 2}{6} + \frac{2 \cdot 6}{6} - (\frac{7}{3} \cdot \frac{2}{2}) = 4$

Step 1.6.5

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

$\frac{5 x \cdot 3 - x \cdot 2}{6} + \frac{2 \cdot 6}{6} - \frac{7 \cdot 2}{3 \cdot 2} = 4$

Step 1.6.6

Reorder the factors of $3 \cdot 2$ .

$\frac{5 x \cdot 3 - x \cdot 2}{6} + \frac{2 \cdot 6}{6} - \frac{7 \cdot 2}{2 \cdot 3} = 4$

Step 1.6.7

Multiply $2$ by $3$ .

$\frac{5 x \cdot 3 - x \cdot 2}{6} + \frac{2 \cdot 6}{6} - \frac{7 \cdot 2}{6} = 4$

Step 1.7

Combine the numerators over the common denominator.

$\frac{5 x \cdot 3 - x \cdot 2 + 2 \cdot 6 - 7 \cdot 2}{6} = 4$

Step 1.8

Simplify each term.

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Step 1.8.1

Multiply $3$ by $5$ .

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

Step 1.8.2

Multiply $2$ by $- 1$ .

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

Step 1.8.3

Multiply $2$ by $6$ .

$\frac{15 x - 2 x + 12 - 7 \cdot 2}{6} = 4$

Step 1.8.4

Multiply $- 7$ by $2$ .

$\frac{15 x - 2 x + 12 - 14}{6} = 4$

Step 1.9

Simplify by adding terms.

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Step 1.9.1

Subtract $2 x$ from $15 x$ .

$\frac{13 x + 12 - 14}{6} = 4$

Step 1.9.2

Subtract $14$ from $12$ .

$\frac{13 x - 2}{6} = 4$

Step 2

Multiply both sides by $6$ .

$\frac{13 x - 2}{6} \cdot 6 = 4 \cdot 6$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $6$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{13 x - 2}{6} \cdot 6 = 4 \cdot 6$

Step 3.1.1.2

Rewrite the expression.

$13 x - 2 = 4 \cdot 6$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $4$ by $6$ .

$13 x - 2 = 24$

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 $2$ to both sides of the equation.

$13 x = 24 + 2$

Step 4.1.2

Add $24$ and $2$ .

$13 x = 26$

Step 4.2

Divide each term in $13 x = 26$ by $13$ and simplify.

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Step 4.2.1

Divide each term in $13 x = 26$ by $13$ .

$\frac{13 x}{13} = \frac{26}{13}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $13$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{13 x}{13} = \frac{26}{13}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{26}{13}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $26$ by $13$ .

$x = 2$