Algebra Examples

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

Step 1

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

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Step 1.1

Rewrite.

$0 + 0 - \frac{1}{2} \cdot (\frac{3}{4} w - 2 + \frac{3}{2} w) = - 3 w$

Step 1.2

Simplify by adding zeros.

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

Step 1.3

Simplify each term.

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Step 1.3.1

Combine $\frac{3}{4}$ and $w$ .

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

Step 1.3.2

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

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

Step 1.4

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

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

Step 1.5

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

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

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

Step 1.5.2

Multiply $2$ by $2$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

Factor $3 w$ out of $3 w + 3 w \cdot 2$ .

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Step 1.7.1.1

Factor $3 w$ out of $3 w$ .

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

Step 1.7.1.2

Factor $3 w$ out of $3 w \cdot 2$ .

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

Step 1.7.1.3

Factor $3 w$ out of $3 w (1) + 3 w (2)$ .

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

Step 1.7.2

Add $1$ and $2$ .

$- \frac{1}{2} \cdot (\frac{3 w \cdot 3}{4} - 2) = - 3 w$

Step 1.7.3

Multiply $3$ by $3$ .

$- \frac{1}{2} \cdot (\frac{9 w}{4} - 2) = - 3 w$

Step 1.8

Apply the distributive property.

$- \frac{1}{2} \cdot \frac{9 w}{4} - \frac{1}{2} \cdot - 2 = - 3 w$

Step 1.9

Multiply $- \frac{1}{2} \cdot \frac{9 w}{4}$ .

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Step 1.9.1

Multiply $\frac{9 w}{4}$ by $\frac{1}{2}$ .

$- \frac{9 w}{4 \cdot 2} - \frac{1}{2} \cdot - 2 = - 3 w$

Step 1.9.2

Multiply $4$ by $2$ .

$- \frac{9 w}{8} - \frac{1}{2} \cdot - 2 = - 3 w$

Step 1.10

Cancel the common factor of $2$ .

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Step 1.10.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{9 w}{8} + \frac{- 1}{2} \cdot - 2 = - 3 w$

Step 1.10.2

Factor $2$ out of $- 2$ .

$- \frac{9 w}{8} + \frac{- 1}{2} \cdot (2 (- 1)) = - 3 w$

Step 1.10.3

Cancel the common factor.

$- \frac{9 w}{8} + \frac{- 1}{2} \cdot (2 \cdot - 1) = - 3 w$

Step 1.10.4

Rewrite the expression.

$- \frac{9 w}{8} - 1 \cdot - 1 = - 3 w$

Step 1.11

Multiply $- 1$ by $- 1$ .

$- \frac{9 w}{8} + 1 = - 3 w$

Step 2

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

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Step 2.1

Add $3 w$ to both sides of the equation.

$- \frac{9 w}{8} + 1 + 3 w = 0$

Step 2.2

To write $3 w$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{9 w}{8} + 3 w \cdot \frac{8}{8} + 1 = 0$

Step 2.3

Combine $3 w$ and $\frac{8}{8}$ .

$- \frac{9 w}{8} + \frac{3 w \cdot 8}{8} + 1 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{- 9 w + 3 w \cdot 8}{8} + 1 = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $3 w$ out of $- 9 w + 3 w \cdot 8$ .

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Step 2.5.1.1

Factor $3 w$ out of $- 9 w$ .

$\frac{3 w (- 3) + 3 w \cdot 8}{8} + 1 = 0$

Step 2.5.1.2

Factor $3 w$ out of $3 w \cdot 8$ .

$\frac{3 w (- 3) + 3 w (8)}{8} + 1 = 0$

Step 2.5.1.3

Factor $3 w$ out of $3 w (- 3) + 3 w (8)$ .

$\frac{3 w (- 3 + 8)}{8} + 1 = 0$

Step 2.5.2

Add $- 3$ and $8$ .

$\frac{3 w \cdot 5}{8} + 1 = 0$

Step 2.5.3

Multiply $5$ by $3$ .

$\frac{15 w}{8} + 1 = 0$

Step 3

Subtract $1$ from both sides of the equation.

$\frac{15 w}{8} = - 1$

Step 4

Multiply both sides of the equation by $\frac{8}{15}$ .

$\frac{8}{15} \cdot \frac{15 w}{8} = \frac{8}{15} \cdot - 1$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{8}{15} \cdot \frac{15 w}{8}$ .

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Step 5.1.1.1

Cancel the common factor of $8$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{8}{15} \cdot \frac{15 w}{8} = \frac{8}{15} \cdot - 1$

Step 5.1.1.1.2

Rewrite the expression.

$\frac{1}{15} (15 w) = \frac{8}{15} \cdot - 1$

Step 5.1.1.2

Cancel the common factor of $15$ .

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Step 5.1.1.2.1

Factor $15$ out of $15 w$ .

$\frac{1}{15} (15 (w)) = \frac{8}{15} \cdot - 1$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{1}{15} (15 w) = \frac{8}{15} \cdot - 1$

Step 5.1.1.2.3

Rewrite the expression.

$w = \frac{8}{15} \cdot - 1$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{8}{15} \cdot - 1$ .

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Step 5.2.1.1

Multiply $\frac{8}{15} \cdot - 1$ .

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Step 5.2.1.1.1

Combine $\frac{8}{15}$ and $- 1$ .

$w = \frac{8 \cdot - 1}{15}$

Step 5.2.1.1.2

Multiply $8$ by $- 1$ .

$w = \frac{- 8}{15}$

Step 5.2.1.2

Move the negative in front of the fraction.

$w = - \frac{8}{15}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{8}{15}$

Decimal Form:

$w = - 0.5\overline{3}$