Algebra Examples

$2 | x - 7 | = x - 8$

Step 1

Divide each term in $2 | x - 7 | = x - 8$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 | x - 7 | = x - 8$ by $2$ .

$\frac{2 | x - 7 |}{2} = \frac{x}{2} + \frac{- 8}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 | x - 7 |}{2} = \frac{x}{2} + \frac{- 8}{2}$

Step 1.2.1.2

Divide $| x - 7 |$ by $1$ .

$| x - 7 | = \frac{x}{2} + \frac{- 8}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $- 8$ by $2$ .

$| x - 7 | = \frac{x}{2} - 4$

Step 2

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

$x - 7 = \pm (\frac{x}{2} - 4)$

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.

$x - 7 = \frac{x}{2} - 4$

Step 3.2

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

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Step 3.2.1

Subtract $\frac{x}{2}$ from both sides of the equation.

$x - 7 - \frac{x}{2} = - 4$

Step 3.2.2

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

$x \cdot \frac{2}{2} - \frac{x}{2} - 7 = - 4$

Step 3.2.3

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

$\frac{x \cdot 2}{2} - \frac{x}{2} - 7 = - 4$

Step 3.2.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - x}{2} - 7 = - 4$

Step 3.2.5

Subtract $x$ from $x \cdot 2$ .

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Step 3.2.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x - x}{2} - 7 = - 4$

Step 3.2.5.2

Subtract $x$ from $2 \cdot x$ .

$\frac{x}{2} - 7 = - 4$

Step 3.3

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

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Step 3.3.1

Add $7$ to both sides of the equation.

$\frac{x}{2} = - 4 + 7$

Step 3.3.2

Add $- 4$ and $7$ .

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

Step 3.4

Multiply both sides of the equation by $2$ .

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

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Cancel the common factor of $2$ .

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Step 3.5.1.1.1

Cancel the common factor.

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

Step 3.5.1.1.2

Rewrite the expression.

$x = 2 \cdot 3$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Multiply $2$ by $3$ .

$x = 6$

Step 3.6

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

$x - 7 = - (\frac{x}{2} - 4)$

Step 3.7

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

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Step 3.7.1

Rewrite.

$x - 7 = 0 + 0 - (\frac{x}{2} - 4)$

Step 3.7.2

Simplify by adding zeros.

$x - 7 = - (\frac{x}{2} - 4)$

Step 3.7.3

Apply the distributive property.

$x - 7 = - \frac{x}{2} - - 4$

Step 3.7.4

Multiply $- 1$ by $- 4$ .

$x - 7 = - \frac{x}{2} + 4$

Step 3.8

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

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Step 3.8.1

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

$x - 7 + \frac{x}{2} = 4$

Step 3.8.2

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

$x \cdot \frac{2}{2} + \frac{x}{2} - 7 = 4$

Step 3.8.3

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

$\frac{x \cdot 2}{2} + \frac{x}{2} - 7 = 4$

Step 3.8.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 + x}{2} - 7 = 4$

Step 3.8.5

Add $x \cdot 2$ and $x$ .

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Step 3.8.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x + x}{2} - 7 = 4$

Step 3.8.5.2

Add $2 \cdot x$ and $x$ .

$\frac{3 \cdot x}{2} - 7 = 4$

$\frac{3 x}{2} - 7 = 4$

Step 3.9

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

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Step 3.9.1

Add $7$ to both sides of the equation.

$\frac{3 x}{2} = 4 + 7$

Step 3.9.2

Add $4$ and $7$ .

$\frac{3 x}{2} = 11$

Step 3.10

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot 11$

Step 3.11

Simplify both sides of the equation.

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Step 3.11.1

Simplify the left side.

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Step 3.11.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Step 3.11.1.1.1

Cancel the common factor of $2$ .

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Step 3.11.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot 11$

Step 3.11.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} \cdot 11$

Step 3.11.1.1.2

Cancel the common factor of $3$ .

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Step 3.11.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} \cdot 11$

Step 3.11.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} \cdot 11$

Step 3.11.1.1.2.3

Rewrite the expression.

$x = \frac{2}{3} \cdot 11$

Step 3.11.2

Simplify the right side.

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Step 3.11.2.1

Multiply $\frac{2}{3} \cdot 11$ .

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Step 3.11.2.1.1

Combine $\frac{2}{3}$ and $11$ .

$x = \frac{2 \cdot 11}{3}$

Step 3.11.2.1.2

Multiply $2$ by $11$ .

$x = \frac{22}{3}$

Step 3.12

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

$x = 6, \frac{22}{3}$

Step 4

Exclude the solutions that do not make $2 | x - 7 | = x - 8$ true.

$No solution$