Algebra Examples

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

Step 1

Simplify the left side.

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Step 1.1

Convert $2 \frac{1}{2}$ to an improper fraction.

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Step 1.1.1

A mixed number is an addition of its whole and fractional parts.

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

Step 1.1.2

Add $2$ and $\frac{1}{2}$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Combine the numerators over the common denominator.

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

Step 1.1.2.4

Simplify the numerator.

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Step 1.1.2.4.1

Multiply $2$ by $2$ .

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

Step 1.1.2.4.2

Add $4$ and $1$ .

$x + \frac{5}{2} = \frac{4 x + 3}{4} - \frac{2 - 3 x}{8}$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\frac{4 x + 3}{4} - \frac{2 - 3 x}{8}$ .

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Step 2.1.1

To write $\frac{4 x + 3}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x + \frac{5}{2} = \frac{4 x + 3}{4} \cdot \frac{2}{2} - \frac{2 - 3 x}{8}$

Step 2.1.2

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

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Step 2.1.2.1

Multiply $\frac{4 x + 3}{4}$ by $\frac{2}{2}$ .

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

Step 2.1.2.2

Multiply $4$ by $2$ .

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

Step 2.1.3

Combine the numerators over the common denominator.

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

Step 2.1.4

Simplify the numerator.

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Step 2.1.4.1

Apply the distributive property.

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

Step 2.1.4.2

Multiply $2$ by $4$ .

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

Step 2.1.4.3

Multiply $3$ by $2$ .

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

Step 2.1.4.4

Apply the distributive property.

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

Step 2.1.4.5

Multiply $- 1$ by $2$ .

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

Step 2.1.4.6

Multiply $- 3$ by $- 1$ .

$x + \frac{5}{2} = \frac{8 x + 6 - 2 + 3 x}{8}$

Step 2.1.4.7

Add $8 x$ and $3 x$ .

$x + \frac{5}{2} = \frac{11 x + 6 - 2}{8}$

Step 2.1.4.8

Subtract $2$ from $6$ .

$x + \frac{5}{2} = \frac{11 x + 4}{8}$

Step 3

Multiply both sides by $8$ .

$(x + \frac{5}{2}) \cdot 8 = \frac{11 x + 4}{8} \cdot 8$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $(x + \frac{5}{2}) \cdot 8$ .

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Step 4.1.1.1

Apply the distributive property.

$x \cdot 8 + \frac{5}{2} \cdot 8 = \frac{11 x + 4}{8} \cdot 8$

Step 4.1.1.2

Move $8$ to the left of $x$ .

$8 \cdot x + \frac{5}{2} \cdot 8 = \frac{11 x + 4}{8} \cdot 8$

Step 4.1.1.3

Cancel the common factor of $2$ .

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Step 4.1.1.3.1

Factor $2$ out of $8$ .

$8 \cdot x + \frac{5}{2} \cdot (2 (4)) = \frac{11 x + 4}{8} \cdot 8$

Step 4.1.1.3.2

Cancel the common factor.

$8 \cdot x + \frac{5}{2} \cdot (2 \cdot 4) = \frac{11 x + 4}{8} \cdot 8$

Step 4.1.1.3.3

Rewrite the expression.

$8 \cdot x + 5 \cdot 4 = \frac{11 x + 4}{8} \cdot 8$

Step 4.1.1.4

Multiply $5$ by $4$ .

$8 x + 20 = \frac{11 x + 4}{8} \cdot 8$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $8$ .

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Step 4.2.1.1

Cancel the common factor.

$8 x + 20 = \frac{11 x + 4}{8} \cdot 8$

Step 4.2.1.2

Rewrite the expression.

$8 x + 20 = 11 x + 4$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Subtract $11 x$ from both sides of the equation.

$8 x + 20 - 11 x = 4$

Step 5.1.2

Subtract $11 x$ from $8 x$ .

$- 3 x + 20 = 4$

Step 5.2

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

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Step 5.2.1

Subtract $20$ from both sides of the equation.

$- 3 x = 4 - 20$

Step 5.2.2

Subtract $20$ from $4$ .

$- 3 x = - 16$

Step 5.3

Divide each term in $- 3 x = - 16$ by $- 3$ and simplify.

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Step 5.3.1

Divide each term in $- 3 x = - 16$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 16}{- 3}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 16}{- 3}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 16}{- 3}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{16}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{16}{3}$

Decimal Form:

$x = 5.\overline{3}$

Mixed Number Form:

$x = 5 \frac{1}{3}$