Algebra Examples

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

Step 1

Simplify $- 2 (3 + x)$ .

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Step 1.1

Apply the distributive property.

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

Step 1.2

Simplify the expression.

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Step 1.2.1

Multiply $- 2$ by $3$ .

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

Step 1.2.2

Reorder $- 6$ and $- 2 x$ .

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

Step 2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

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

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 3.2

Multiply both sides by $2$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Simplify the numerator.

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Step 3.3.1.1.1.1

Apply the distributive property.

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

Step 3.3.1.1.1.2

Multiply $- 1$ by $1$ .

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

Step 3.3.1.1.1.3

Multiply $- - x$ .

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Step 3.3.1.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.1.1.3.2

Multiply $x$ by $1$ .

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

Step 3.3.1.1.1.4

Subtract $1$ from $2$ .

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

Step 3.3.1.1.2

Reduce the expression by cancelling the common factors.

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Step 3.3.1.1.2.1

Cancel the common factor of $2$ .

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Step 3.3.1.1.2.1.1

Cancel the common factor.

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

Step 3.3.1.1.2.1.2

Rewrite the expression.

$1 + x = (- 2 x - 6) \cdot 2$

Step 3.3.1.1.2.2

Reorder $1$ and $x$ .

$x + 1 = (- 2 x - 6) \cdot 2$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $(- 2 x - 6) \cdot 2$ .

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Step 3.3.2.1.1

Apply the distributive property.

$x + 1 = - 2 x \cdot 2 - 6 \cdot 2$

Step 3.3.2.1.2

Multiply.

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Step 3.3.2.1.2.1

Multiply $2$ by $- 2$ .

$x + 1 = - 4 x - 6 \cdot 2$

Step 3.3.2.1.2.2

Multiply $- 6$ by $2$ .

$x + 1 = - 4 x - 12$

Step 3.4

Solve for $x$ .

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Step 3.4.1

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

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Step 3.4.1.1

Add $4 x$ to both sides of the equation.

$x + 1 + 4 x = - 12$

Step 3.4.1.2

Add $x$ and $4 x$ .

$5 x + 1 = - 12$

Step 3.4.2

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

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Step 3.4.2.1

Subtract $1$ from both sides of the equation.

$5 x = - 12 - 1$

Step 3.4.2.2

Subtract $1$ from $- 12$ .

$5 x = - 13$

Step 3.4.3

Divide each term in $5 x = - 13$ by $5$ and simplify.

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Step 3.4.3.1

Divide each term in $5 x = - 13$ by $5$ .

$\frac{5 x}{5} = \frac{- 13}{5}$

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Cancel the common factor of $5$ .

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Step 3.4.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 13}{5}$

Step 3.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 13}{5}$

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Move the negative in front of the fraction.

$x = - \frac{13}{5}$

Step 3.5

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.6

Multiply both sides by $2$ .

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

Step 3.7

Simplify.

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Step 3.7.1

Simplify the left side.

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Step 3.7.1.1

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

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Step 3.7.1.1.1

Simplify the numerator.

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Step 3.7.1.1.1.1

Apply the distributive property.

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

Step 3.7.1.1.1.2

Multiply $- 1$ by $1$ .

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

Step 3.7.1.1.1.3

Multiply $- - x$ .

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Step 3.7.1.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.7.1.1.1.3.2

Multiply $x$ by $1$ .

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

Step 3.7.1.1.1.4

Subtract $1$ from $2$ .

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

Step 3.7.1.1.2

Reduce the expression by cancelling the common factors.

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Step 3.7.1.1.2.1

Cancel the common factor of $2$ .

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Step 3.7.1.1.2.1.1

Cancel the common factor.

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

Step 3.7.1.1.2.1.2

Rewrite the expression.

$1 + x = - (- 2 x - 6) \cdot 2$

Step 3.7.1.1.2.2

Reorder $1$ and $x$ .

$x + 1 = - (- 2 x - 6) \cdot 2$

Step 3.7.2

Simplify the right side.

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Step 3.7.2.1

Simplify $- (- 2 x - 6) \cdot 2$ .

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Step 3.7.2.1.1

Apply the distributive property.

$x + 1 = (- (- 2 x) - - 6) \cdot 2$

Step 3.7.2.1.2

Multiply.

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Step 3.7.2.1.2.1

Multiply $- 2$ by $- 1$ .

$x + 1 = (2 x - - 6) \cdot 2$

Step 3.7.2.1.2.2

Multiply $- 1$ by $- 6$ .

$x + 1 = (2 x + 6) \cdot 2$

Step 3.7.2.1.3

Apply the distributive property.

$x + 1 = 2 x \cdot 2 + 6 \cdot 2$

Step 3.7.2.1.4

Multiply.

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Step 3.7.2.1.4.1

Multiply $2$ by $2$ .

$x + 1 = 4 x + 6 \cdot 2$

Step 3.7.2.1.4.2

Multiply $6$ by $2$ .

$x + 1 = 4 x + 12$

Step 3.8

Solve for $x$ .

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Step 3.8.1

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

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Step 3.8.1.1

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

$x + 1 - 4 x = 12$

Step 3.8.1.2

Subtract $4 x$ from $x$ .

$- 3 x + 1 = 12$

Step 3.8.2

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

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Step 3.8.2.1

Subtract $1$ from both sides of the equation.

$- 3 x = 12 - 1$

Step 3.8.2.2

Subtract $1$ from $12$ .

$- 3 x = 11$

Step 3.8.3

Divide each term in $- 3 x = 11$ by $- 3$ and simplify.

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Step 3.8.3.1

Divide each term in $- 3 x = 11$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{11}{- 3}$

Step 3.8.3.2

Simplify the left side.

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Step 3.8.3.2.1

Cancel the common factor of $- 3$ .

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Step 3.8.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{11}{- 3}$

Step 3.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{- 3}$

Step 3.8.3.3

Simplify the right side.

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Step 3.8.3.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{3}$

Step 3.9

The complete solution is the result of both the positive and negative portions of the solution.

$x = - \frac{13}{5}, - \frac{11}{3}$

Step 4

Exclude the solutions that do not make $| \frac{2 - (1 - x)}{2} | = - 2 (3 + x)$ true.

$x = - \frac{11}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{11}{3}$

Decimal Form:

$x = - 3.\overline{6}$

Mixed Number Form:

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