Precalculus Examples

$f (x) = | - x + 5 | + 2 x + 3$

Step 1

Set $| - x + 5 | + 2 x + 3$ equal to $0$ .

$| - x + 5 | + 2 x + 3 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Move all terms not containing $| - x + 5 |$ to the right side of the equation.

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Step 2.1.1

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

$| - x + 5 | + 3 = - 2 x$

Step 2.1.2

Subtract $3$ from both sides of the equation.

$| - x + 5 | = - 2 x - 3$

Step 2.2

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

$- x + 5 = \pm (- 2 x - 3)$

Step 2.3

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

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Step 2.3.1

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

$- x + 5 = - 2 x - 3$

Step 2.3.2

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

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Step 2.3.2.1

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

$- x + 5 + 2 x = - 3$

Step 2.3.2.2

Add $- x$ and $2 x$ .

$x + 5 = - 3$

Step 2.3.3

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

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Step 2.3.3.1

Subtract $5$ from both sides of the equation.

$x = - 3 - 5$

Step 2.3.3.2

Subtract $5$ from $- 3$ .

$x = - 8$

Step 2.3.4

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

$- x + 5 = - (- 2 x - 3)$

Step 2.3.5

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

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Step 2.3.5.1

Rewrite.

$- x + 5 = 0 + 0 - (- 2 x - 3)$

Step 2.3.5.2

Simplify by adding zeros.

$- x + 5 = - (- 2 x - 3)$

Step 2.3.5.3

Apply the distributive property.

$- x + 5 = - (- 2 x) - - 3$

Step 2.3.5.4

Multiply.

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Step 2.3.5.4.1

Multiply $- 2$ by $- 1$ .

$- x + 5 = 2 x - - 3$

Step 2.3.5.4.2

Multiply $- 1$ by $- 3$ .

$- x + 5 = 2 x + 3$

Step 2.3.6

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

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Step 2.3.6.1

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

$- x + 5 - 2 x = 3$

Step 2.3.6.2

Subtract $2 x$ from $- x$ .

$- 3 x + 5 = 3$

Step 2.3.7

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

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Step 2.3.7.1

Subtract $5$ from both sides of the equation.

$- 3 x = 3 - 5$

Step 2.3.7.2

Subtract $5$ from $3$ .

$- 3 x = - 2$

Step 2.3.8

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

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Step 2.3.8.1

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

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

Step 2.3.8.2

Simplify the left side.

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Step 2.3.8.2.1

Cancel the common factor of $- 3$ .

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Step 2.3.8.2.1.1

Cancel the common factor.

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

Step 2.3.8.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.8.3

Simplify the right side.

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Step 2.3.8.3.1

Dividing two negative values results in a positive value.

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

Step 2.3.9

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

$x = - 8, \frac{2}{3}$

Step 2.4

Exclude the solutions that do not make $| - x + 5 | + 2 x + 3 = 0$ true.

$x = - 8$

Step 3