Algebra Examples

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

Step 1

Simplify $- 2 (- x - \frac{1}{4}) - 8$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Multiply $- 1$ by $- 2$ .

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

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

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

Step 1.1.3.2

Factor $2$ out of $- 2$ .

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

Step 1.1.3.3

Factor $2$ out of $4$ .

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

Step 1.1.3.4

Cancel the common factor.

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

Step 1.1.3.5

Rewrite the expression.

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

Step 1.1.4

Simplify each term.

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Step 1.1.4.1

Move the negative in front of the fraction.

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

Step 1.1.4.2

Multiply $- 1 (- \frac{1}{2})$ .

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Step 1.1.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.1.4.2.2

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

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

Step 1.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.3

Combine $- 8$ and $\frac{2}{2}$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 8$ by $2$ .

$2 x + \frac{1 - 16}{2} = - \frac{3}{2} \cdot (- 2 x + 1) - 1$

Step 1.5.2

Subtract $16$ from $1$ .

$2 x + \frac{- 15}{2} = - \frac{3}{2} \cdot (- 2 x + 1) - 1$

Step 1.6

Move the negative in front of the fraction.

$2 x - \frac{15}{2} = - \frac{3}{2} \cdot (- 2 x + 1) - 1$

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$2 x - \frac{15}{2} = - \frac{3}{2} (- 2 x) - \frac{3}{2} \cdot 1 - 1$

Step 2.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$2 x - \frac{15}{2} = \frac{- 3}{2} (- 2 x) - \frac{3}{2} \cdot 1 - 1$

Step 2.1.2.2

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

$2 x - \frac{15}{2} = \frac{- 3}{2} (2 (- x)) - \frac{3}{2} \cdot 1 - 1$

Step 2.1.2.3

Cancel the common factor.

$2 x - \frac{15}{2} = \frac{- 3}{2} (2 (- x)) - \frac{3}{2} \cdot 1 - 1$

Step 2.1.2.4

Rewrite the expression.

$2 x - \frac{15}{2} = - 3 (- x) - \frac{3}{2} \cdot 1 - 1$

Step 2.1.3

Multiply $- 1$ by $- 3$ .

$2 x - \frac{15}{2} = 3 x - \frac{3}{2} \cdot 1 - 1$

Step 2.1.4

Multiply $- 1$ by $1$ .

$2 x - \frac{15}{2} = 3 x - \frac{3}{2} - 1$

Step 2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x - \frac{15}{2} = 3 x - \frac{3}{2} - 1 \cdot \frac{2}{2}$

Step 2.3

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

$2 x - \frac{15}{2} = 3 x - \frac{3}{2} + \frac{- 1 \cdot 2}{2}$

Step 2.4

Combine the numerators over the common denominator.

$2 x - \frac{15}{2} = 3 x + \frac{- 3 - 1 \cdot 2}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $2$ .

$2 x - \frac{15}{2} = 3 x + \frac{- 3 - 2}{2}$

Step 2.5.2

Subtract $2$ from $- 3$ .

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

Step 2.6

Move the negative in front of the fraction.

$2 x - \frac{15}{2} = 3 x - \frac{5}{2}$

Step 3

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

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Step 3.1

Subtract $3 x$ from both sides of the equation.

$2 x - \frac{15}{2} - 3 x = - \frac{5}{2}$

Step 3.2

Subtract $3 x$ from $2 x$ .

$- x - \frac{15}{2} = - \frac{5}{2}$

Step 4

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

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Step 4.1

Add $\frac{15}{2}$ to both sides of the equation.

$- x = - \frac{5}{2} + \frac{15}{2}$

Step 4.2

Combine the numerators over the common denominator.

$- x = \frac{- 5 + 15}{2}$

Step 4.3

Add $- 5$ and $15$ .

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

Step 4.4

Divide $10$ by $2$ .

$- x = 5$

Step 5

Divide each term in $- x = 5$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- x = 5$ by $- 1$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2

Divide $x$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $5$ by $- 1$ .

$x = - 5$