Calculus Examples

$| x + 5 | = 2 x - 7$

Step 1

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

$x + 5 = \pm (2 x - 7)$

Step 2

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

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Step 2.1

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

$x + 5 = 2 x - 7$

Step 2.2

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

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Step 2.2.1

Subtract $2 x$ from both sides of the equation.

$x + 5 - 2 x = - 7$

Step 2.2.2

Subtract $2 x$ from $x$ .

$- x + 5 = - 7$

Step 2.3

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

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Step 2.3.1

Subtract $5$ from both sides of the equation.

$- x = - 7 - 5$

Step 2.3.2

Subtract $5$ from $- 7$ .

$- x = - 12$

Step 2.4

Divide each term in $- x = - 12$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- x = - 12$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 12}{- 1}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 12}{- 1}$

Step 2.4.2.2

Divide $x$ by $1$ .

$x = \frac{- 12}{- 1}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $- 12$ by $- 1$ .

$x = 12$

Step 2.5

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

$x + 5 = - (2 x - 7)$

Step 2.6

Simplify $- (2 x - 7)$ .

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Step 2.6.1

Rewrite.

$x + 5 = 0 + 0 - (2 x - 7)$

Step 2.6.2

Simplify by adding zeros.

$x + 5 = - (2 x - 7)$

Step 2.6.3

Apply the distributive property.

$x + 5 = - (2 x) - - 7$

Step 2.6.4

Multiply.

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Step 2.6.4.1

Multiply $2$ by $- 1$ .

$x + 5 = - 2 x - - 7$

Step 2.6.4.2

Multiply $- 1$ by $- 7$ .

$x + 5 = - 2 x + 7$

Step 2.7

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

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Step 2.7.1

Add $2 x$ to both sides of the equation.

$x + 5 + 2 x = 7$

Step 2.7.2

Add $x$ and $2 x$ .

$3 x + 5 = 7$

Step 2.8

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

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Step 2.8.1

Subtract $5$ from both sides of the equation.

$3 x = 7 - 5$

Step 2.8.2

Subtract $5$ from $7$ .

$3 x = 2$

Step 2.9

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 2.9.1

Divide each term in $3 x = 2$ by $3$ .

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

Step 2.9.2

Simplify the left side.

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Step 2.9.2.1

Cancel the common factor of $3$ .

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Step 2.9.2.1.1

Cancel the common factor.

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

Step 2.9.2.1.2

Divide $x$ by $1$ .

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

Step 2.10

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

$x = 12, \frac{2}{3}$

Step 3

Exclude the solutions that do not make $| x + 5 | = 2 x - 7$ true.

$x = 12$