Algebra Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Move $7$ to the left of $x$ .

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

Step 1.1.3

Apply the distributive property.

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

Step 1.1.4

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

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

Step 1.1.5

Multiply $\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 1.1.5.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

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

Step 1.1.5.2

Multiply $2$ by $3$ .

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

Step 1.2

Simplify terms.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

Add $- 7 x$ and $x$ .

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

Step 1.2.3

Cancel the common factor of $- 6$ and $2$ .

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Step 1.2.3.1

Factor $2$ out of $- 6 x$ .

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

Step 1.2.3.2

Cancel the common factors.

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Step 1.2.3.2.1

Factor $2$ out of $2$ .

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

Step 1.2.3.2.2

Cancel the common factor.

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

Step 1.2.3.2.3

Rewrite the expression.

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

Step 1.2.3.2.4

Divide $- 3 x$ by $1$ .

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

Step 1.3

To write $\frac{2}{5}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 1.4

To write $\frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.5

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{2}{5}$ by $\frac{6}{6}$ .

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

Step 1.5.2

Multiply $5$ by $6$ .

$- 3 x + \frac{2 \cdot 6}{30} + \frac{1}{6} \cdot \frac{5}{5} = - \frac{5}{2} + x$

Step 1.5.3

Multiply $\frac{1}{6}$ by $\frac{5}{5}$ .

$- 3 x + \frac{2 \cdot 6}{30} + \frac{5}{6 \cdot 5} = - \frac{5}{2} + x$

Step 1.5.4

Multiply $6$ by $5$ .

$- 3 x + \frac{2 \cdot 6}{30} + \frac{5}{30} = - \frac{5}{2} + x$

Step 1.6

Combine the numerators over the common denominator.

$- 3 x + \frac{2 \cdot 6 + 5}{30} = - \frac{5}{2} + x$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $2$ by $6$ .

$- 3 x + \frac{12 + 5}{30} = - \frac{5}{2} + x$

Step 1.7.2

Add $12$ and $5$ .

$- 3 x + \frac{17}{30} = - \frac{5}{2} + x$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the equation.

$- 3 x + \frac{17}{30} - x = - \frac{5}{2}$

Step 2.2

Subtract $x$ from $- 3 x$ .

$- 4 x + \frac{17}{30} = - \frac{5}{2}$

Step 3

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

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Step 3.1

Subtract $\frac{17}{30}$ from both sides of the equation.

$- 4 x = - \frac{5}{2} - \frac{17}{30}$

Step 3.2

To write $- \frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$- 4 x = - \frac{5}{2} \cdot \frac{15}{15} - \frac{17}{30}$

Step 3.3

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{5}{2}$ by $\frac{15}{15}$ .

$- 4 x = - \frac{5 \cdot 15}{2 \cdot 15} - \frac{17}{30}$

Step 3.3.2

Multiply $2$ by $15$ .

$- 4 x = - \frac{5 \cdot 15}{30} - \frac{17}{30}$

Step 3.4

Combine the numerators over the common denominator.

$- 4 x = \frac{- 5 \cdot 15 - 17}{30}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 5$ by $15$ .

$- 4 x = \frac{- 75 - 17}{30}$

Step 3.5.2

Subtract $17$ from $- 75$ .

$- 4 x = \frac{- 92}{30}$

Step 3.6

Cancel the common factor of $- 92$ and $30$ .

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Step 3.6.1

Factor $2$ out of $- 92$ .

$- 4 x = \frac{2 (- 46)}{30}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $2$ out of $30$ .

$- 4 x = \frac{2 \cdot - 46}{2 \cdot 15}$

Step 3.6.2.2

Cancel the common factor.

$- 4 x = \frac{2 \cdot - 46}{2 \cdot 15}$

Step 3.6.2.3

Rewrite the expression.

$- 4 x = \frac{- 46}{15}$

Step 3.7

Move the negative in front of the fraction.

$- 4 x = - \frac{46}{15}$

Step 4

Divide each term in $- 4 x = - \frac{46}{15}$ by $- 4$ and simplify.

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Step 4.1

Divide each term in $- 4 x = - \frac{46}{15}$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- \frac{46}{15}}{- 4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- \frac{46}{15}}{- 4}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{46}{15}}{- 4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{46}{15} \cdot \frac{1}{- 4}$

Step 4.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.1

Move the leading negative in $- \frac{46}{15}$ into the numerator.

$x = \frac{- 46}{15} \cdot \frac{1}{- 4}$

Step 4.3.2.2

Factor $2$ out of $- 46$ .

$x = \frac{2 (- 23)}{15} \cdot \frac{1}{- 4}$

Step 4.3.2.3

Factor $2$ out of $- 4$ .

$x = \frac{2 \cdot - 23}{15} \cdot \frac{1}{2 \cdot - 2}$

Step 4.3.2.4

Cancel the common factor.

$x = \frac{2 \cdot - 23}{15} \cdot \frac{1}{2 \cdot - 2}$

Step 4.3.2.5

Rewrite the expression.

$x = \frac{- 23}{15} \cdot \frac{1}{- 2}$

Step 4.3.3

Multiply $\frac{- 23}{15}$ by $\frac{1}{- 2}$ .

$x = \frac{- 23}{15 \cdot - 2}$

Step 4.3.4

Multiply $15$ by $- 2$ .

$x = \frac{- 23}{- 30}$

Step 4.3.5

Dividing two negative values results in a positive value.

$x = \frac{23}{30}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{23}{30}$

Decimal Form:

$x = 0.7\overline{6}$