Algebra Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

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

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

Step 1.1.4

Combine $\frac{1}{2}$ and $3$ .

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

Step 1.2

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

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

Step 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 1.3.1

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

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

Step 1.3.2

Multiply $2$ by $2$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Add $x$ and $x \cdot 2$ .

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Step 1.5.1

Reorder $x$ and $2$ .

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

Step 1.5.2

Add $x$ and $2 \cdot x$ .

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

Step 2

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

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Step 2.1

Subtract $\frac{3}{2}$ from both sides of the equation.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $5$ by $2$ .

$\frac{3 x}{4} = \frac{10 - 3}{2}$

Step 2.5.2

Subtract $3$ from $10$ .

$\frac{3 x}{4} = \frac{7}{2}$

Step 3

Multiply both sides of the equation by $\frac{4}{3}$ .

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

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{4}{3} \cdot \frac{3 x}{4}$ .

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Step 4.1.1.1

Cancel the common factor of $4$ .

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Step 4.1.1.1.1

Cancel the common factor.

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

Step 4.1.1.1.2

Rewrite the expression.

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

Step 4.1.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.2.1

Factor $3$ out of $3 x$ .

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

Step 4.1.1.2.2

Cancel the common factor.

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

Step 4.1.1.2.3

Rewrite the expression.

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{4}{3} \cdot \frac{7}{2}$ .

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{3} \cdot \frac{7}{2}$

Step 4.2.1.1.2

Cancel the common factor.

$x = \frac{2 \cdot 2}{3} \cdot \frac{7}{2}$

Step 4.2.1.1.3

Rewrite the expression.

$x = \frac{2}{3} \cdot 7$

Step 4.2.1.2

Combine $\frac{2}{3}$ and $7$ .

$x = \frac{2 \cdot 7}{3}$

Step 4.2.1.3

Multiply $2$ by $7$ .

$x = \frac{14}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{14}{3}$

Decimal Form:

$x = 4.\overline{6}$

Mixed Number Form:

$x = 4 \frac{2}{3}$