Algebra Examples

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

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

Simplify $3 x + \frac{1}{2} \cdot (- 6 x + 1)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.1

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

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

Step 2.1.2.2

Cancel the common factor.

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

Step 2.1.2.3

Rewrite the expression.

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

Step 2.1.3

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

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $3 x$ from $3 x$ .

$0 + \frac{1}{2} = - \frac{1}{2} \cdot (- x + 8)$

Step 2.2.2

Add $0$ and $\frac{1}{2}$ .

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

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

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

Step 3.3.2

Factor $2$ out of $8$ .

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Multiply $- 1$ by $4$ .

$\frac{1}{2} = \frac{x}{2} - 4$

Step 4

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{x}{2} - 4 = \frac{1}{2}$

Step 5

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

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Step 5.1

Add $4$ to both sides of the equation.

$\frac{x}{2} = \frac{1}{2} + 4$

Step 5.2

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

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

Step 5.3

Combine $4$ and $\frac{2}{2}$ .

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

Step 5.4

Combine the numerators over the common denominator.

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

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $4$ by $2$ .

$\frac{x}{2} = \frac{1 + 8}{2}$

Step 5.5.2

Add $1$ and $8$ .

$\frac{x}{2} = \frac{9}{2}$

Step 6

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = 9$