Algebra Examples

$\frac{1}{2} x - 20 - x = 3 (2 x - 1) - (2 x - 1)$

Step 1

Simplify $\frac{1}{2} x - 20 - x$ .

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Step 1.1

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

$\frac{x}{2} - 20 - x = 3 (2 x - 1) - (2 x - 1)$

Step 1.2

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

$\frac{x}{2} - x \cdot \frac{2}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.3

Simplify terms.

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Step 1.3.1

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

$\frac{x}{2} + \frac{- x \cdot 2}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{x - x \cdot 2}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.4

Simplify each term.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

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

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Step 1.4.1.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} - x \cdot 2}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.4.1.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 - x \cdot 2}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.4.1.1.3

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

$\frac{x \cdot 1 + x (- 1 \cdot 2)}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.4.1.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 \cdot 2)$ .

$\frac{x (1 - 1 \cdot 2)}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.4.1.2

Multiply $- 1$ by $2$ .

$\frac{x (1 - 2)}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.4.1.3

Subtract $2$ from $1$ .

$\frac{x \cdot - 1}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.4.2

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 1.4.3

Move the negative in front of the fraction.

$- \frac{x}{2} - 20 = 3 (2 x - 1) - (2 x - 1)$

Step 2

Simplify $3 (2 x - 1) - (2 x - 1)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- \frac{x}{2} - 20 = 3 (2 x) + 3 \cdot - 1 - (2 x - 1)$

Step 2.1.2

Multiply $2$ by $3$ .

$- \frac{x}{2} - 20 = 6 x + 3 \cdot - 1 - (2 x - 1)$

Step 2.1.3

Multiply $3$ by $- 1$ .

$- \frac{x}{2} - 20 = 6 x - 3 - (2 x - 1)$

Step 2.1.4

Apply the distributive property.

$- \frac{x}{2} - 20 = 6 x - 3 - (2 x) - - 1$

Step 2.1.5

Multiply $2$ by $- 1$ .

$- \frac{x}{2} - 20 = 6 x - 3 - 2 x - - 1$

Step 2.1.6

Multiply $- 1$ by $- 1$ .

$- \frac{x}{2} - 20 = 6 x - 3 - 2 x + 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $2 x$ from $6 x$ .

$- \frac{x}{2} - 20 = 4 x - 3 + 1$

Step 2.2.2

Add $- 3$ and $1$ .

$- \frac{x}{2} - 20 = 4 x - 2$

Step 3

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

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Step 3.1

Subtract $4 x$ from both sides of the equation.

$- \frac{x}{2} - 20 - 4 x = - 2$

Step 3.2

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

$- \frac{x}{2} - 4 x \cdot \frac{2}{2} - 20 = - 2$

Step 3.3

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

$- \frac{x}{2} + \frac{- 4 x \cdot 2}{2} - 20 = - 2$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- x - 4 x \cdot 2}{2} - 20 = - 2$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Factor $x$ out of $- x - 4 x \cdot 2$ .

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Step 3.5.1.1.1

Factor $x$ out of $- x$ .

$\frac{x \cdot - 1 - 4 x \cdot 2}{2} - 20 = - 2$

Step 3.5.1.1.2

Factor $x$ out of $- 4 x \cdot 2$ .

$\frac{x \cdot - 1 + x (- 4 \cdot 2)}{2} - 20 = - 2$

Step 3.5.1.1.3

Factor $x$ out of $x \cdot - 1 + x (- 4 \cdot 2)$ .

$\frac{x (- 1 - 4 \cdot 2)}{2} - 20 = - 2$

Step 3.5.1.2

Multiply $- 4$ by $2$ .

$\frac{x (- 1 - 8)}{2} - 20 = - 2$

Step 3.5.1.3

Subtract $8$ from $- 1$ .

$\frac{x \cdot - 9}{2} - 20 = - 2$

Step 3.5.2

Move $- 9$ to the left of $x$ .

$\frac{- 9 \cdot x}{2} - 20 = - 2$

Step 3.5.3

Move the negative in front of the fraction.

$- \frac{9 x}{2} - 20 = - 2$

Step 4

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

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Step 4.1

Add $20$ to both sides of the equation.

$- \frac{9 x}{2} = - 2 + 20$

Step 4.2

Add $- 2$ and $20$ .

$- \frac{9 x}{2} = 18$

Step 5

Multiply both sides of the equation by $- \frac{2}{9}$ .

$- \frac{2}{9} (- \frac{9 x}{2}) = - \frac{2}{9} \cdot 18$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $- \frac{2}{9} (- \frac{9 x}{2})$ .

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Step 6.1.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1.1

Move the leading negative in $- \frac{2}{9}$ into the numerator.

$\frac{- 2}{9} (- \frac{9 x}{2}) = - \frac{2}{9} \cdot 18$

Step 6.1.1.1.2

Move the leading negative in $- \frac{9 x}{2}$ into the numerator.

$\frac{- 2}{9} \cdot \frac{- 9 x}{2} = - \frac{2}{9} \cdot 18$

Step 6.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{9} \cdot \frac{- 9 x}{2} = - \frac{2}{9} \cdot 18$

Step 6.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{9} \cdot \frac{- 9 x}{2} = - \frac{2}{9} \cdot 18$

Step 6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{9} (- 9 x) = - \frac{2}{9} \cdot 18$

Step 6.1.1.2

Cancel the common factor of $9$ .

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Step 6.1.1.2.1

Factor $9$ out of $- 9 x$ .

$\frac{- 1}{9} (9 (- x)) = - \frac{2}{9} \cdot 18$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{9} (9 (- x)) = - \frac{2}{9} \cdot 18$

Step 6.1.1.2.3

Rewrite the expression.

$- - x = - \frac{2}{9} \cdot 18$

Step 6.1.1.3

Multiply.

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Step 6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{2}{9} \cdot 18$

Step 6.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{2}{9} \cdot 18$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{2}{9} \cdot 18$ .

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Step 6.2.1.1

Cancel the common factor of $9$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{2}{9}$ into the numerator.

$x = \frac{- 2}{9} \cdot 18$

Step 6.2.1.1.2

Factor $9$ out of $18$ .

$x = \frac{- 2}{9} \cdot (9 (2))$

Step 6.2.1.1.3

Cancel the common factor.

$x = \frac{- 2}{9} \cdot (9 \cdot 2)$

Step 6.2.1.1.4

Rewrite the expression.

$x = - 2 \cdot 2$

Step 6.2.1.2

Multiply $- 2$ by $2$ .

$x = - 4$