Algebra Examples

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

Step 1

Subtract $| 3 - x |$ from both sides of the equation.

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

Step 2

Divide each term in $- | x + 2 | = 5 - | 3 - x |$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- | x + 2 | = 5 - | 3 - x |$ by $- 1$ .

$\frac{- | x + 2 |}{- 1} = \frac{5}{- 1} + \frac{- | 3 - x |}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{| x + 2 |}{1} = \frac{5}{- 1} + \frac{- | 3 - x |}{- 1}$

Step 2.2.2

Divide $| x + 2 |$ by $1$ .

$| x + 2 | = \frac{5}{- 1} + \frac{- | 3 - x |}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Divide $5$ by $- 1$ .

$| x + 2 | = - 5 + \frac{- | 3 - x |}{- 1}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$| x + 2 | = - 5 + \frac{| 3 - x |}{1}$

Step 2.3.1.3

Divide $| 3 - x |$ by $1$ .

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

Step 3

Solve for $| 3 - x |$ .

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Step 3.1

Rewrite the equation as $- 5 + | 3 - x | = | x + 2 |$ .

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

Step 3.2

Add $5$ to both sides of the equation.

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

Step 4

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

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

Step 5

The result consists of both the positive and negative portions of the $\pm$ .

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

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

Step 6

Solve $3 - x = | x + 2 | + 5$ for $x$ .

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Step 6.1

Solve for $| x + 2 |$ .

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Step 6.1.1

Rewrite the equation as $| x + 2 | + 5 = 3 - x$ .

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

Step 6.1.2

Move all terms not containing $| x + 2 |$ to the right side of the equation.

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Step 6.1.2.1

Subtract $5$ from both sides of the equation.

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

Step 6.1.2.2

Subtract $5$ from $3$ .

$| x + 2 | = - x - 2$

Step 6.2

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

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

Step 6.3

The result consists of both the positive and negative portions of the $\pm$ .

$x + 2 = - x - 2$

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

Step 6.4

Solve $x + 2 = - x - 2$ for $x$ .

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Step 6.4.1

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

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Step 6.4.1.1

Add $x$ to both sides of the equation.

$x + 2 + x = - 2$

Step 6.4.1.2

Add $x$ and $x$ .

$2 x + 2 = - 2$

Step 6.4.2

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

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Step 6.4.2.1

Subtract $2$ from both sides of the equation.

$2 x = - 2 - 2$

Step 6.4.2.2

Subtract $2$ from $- 2$ .

$2 x = - 4$

Step 6.4.3

Divide each term in $2 x = - 4$ by $2$ and simplify.

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Step 6.4.3.1

Divide each term in $2 x = - 4$ by $2$ .

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

Step 6.4.3.2

Simplify the left side.

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Step 6.4.3.2.1

Cancel the common factor of $2$ .

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Step 6.4.3.2.1.1

Cancel the common factor.

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

Step 6.4.3.2.1.2

Divide $x$ by $1$ .

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

Step 6.4.3.3

Simplify the right side.

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Step 6.4.3.3.1

Divide $- 4$ by $2$ .

$x = - 2$

Step 6.5

Solve $x + 2 = - (- x - 2)$ for $x$ .

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Step 6.5.1

Simplify $- (- x - 2)$ .

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Step 6.5.1.1

Rewrite.

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

Step 6.5.1.2

Simplify by adding zeros.

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

Step 6.5.1.3

Apply the distributive property.

$x + 2 = - - x - - 2$

Step 6.5.1.4

Multiply $- - x$ .

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Step 6.5.1.4.1

Multiply $- 1$ by $- 1$ .

$x + 2 = 1 x - - 2$

Step 6.5.1.4.2

Multiply $x$ by $1$ .

$x + 2 = x - - 2$

Step 6.5.1.5

Multiply $- 1$ by $- 2$ .

$x + 2 = x + 2$

Step 6.5.2

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

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Step 6.5.2.1

Subtract $x$ from both sides of the equation.

$x + 2 - x = 2$

Step 6.5.2.2

Combine the opposite terms in $x + 2 - x$ .

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Step 6.5.2.2.1

Subtract $x$ from $x$ .

$0 + 2 = 2$

Step 6.5.2.2.2

Add $0$ and $2$ .

$2 = 2$

Step 6.5.3

Since $2 = 2$ , the equation will always be true.

Always true

Step 6.6

Consolidate the solutions.

$x = - 2$

Step 7

Solve $3 - x = - (| x + 2 | + 5)$ for $x$ .

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Step 7.1

Solve for $| x + 2 |$ .

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Step 7.1.1

Rewrite the equation as $- (| x + 2 | + 5) = 3 - x$ .

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

Step 7.1.2

Simplify $- (| x + 2 | + 5)$ .

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Step 7.1.2.1

Apply the distributive property.

$- | x + 2 | - 1 \cdot 5 = 3 - x$

Step 7.1.2.2

Multiply $- 1$ by $5$ .

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

Step 7.1.3

Move all terms not containing $| x + 2 |$ to the right side of the equation.

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Step 7.1.3.1

Add $5$ to both sides of the equation.

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

Step 7.1.3.2

Add $3$ and $5$ .

$- | x + 2 | = - x + 8$

Step 7.1.4

Divide each term in $- | x + 2 | = - x + 8$ by $- 1$ and simplify.

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Step 7.1.4.1

Divide each term in $- | x + 2 | = - x + 8$ by $- 1$ .

$\frac{- | x + 2 |}{- 1} = \frac{- x}{- 1} + \frac{8}{- 1}$

Step 7.1.4.2

Simplify the left side.

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Step 7.1.4.2.1

Dividing two negative values results in a positive value.

$\frac{| x + 2 |}{1} = \frac{- x}{- 1} + \frac{8}{- 1}$

Step 7.1.4.2.2

Divide $| x + 2 |$ by $1$ .

$| x + 2 | = \frac{- x}{- 1} + \frac{8}{- 1}$

Step 7.1.4.3

Simplify the right side.

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Step 7.1.4.3.1

Simplify each term.

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Step 7.1.4.3.1.1

Dividing two negative values results in a positive value.

$| x + 2 | = \frac{x}{1} + \frac{8}{- 1}$

Step 7.1.4.3.1.2

Divide $x$ by $1$ .

$| x + 2 | = x + \frac{8}{- 1}$

Step 7.1.4.3.1.3

Divide $8$ by $- 1$ .

$| x + 2 | = x - 8$

Step 7.2

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

$x + 2 = \pm (x - 8)$

Step 7.3

The result consists of both the positive and negative portions of the $\pm$ .

$x + 2 = x - 8$

$x + 2 = - (x - 8)$

Step 7.4

Solve $x + 2 = x - 8$ for $x$ .

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Step 7.4.1

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

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Step 7.4.1.1

Subtract $x$ from both sides of the equation.

$x + 2 - x = - 8$

Step 7.4.1.2

Combine the opposite terms in $x + 2 - x$ .

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Step 7.4.1.2.1

Subtract $x$ from $x$ .

$0 + 2 = - 8$

Step 7.4.1.2.2

Add $0$ and $2$ .

$2 = - 8$

Step 7.4.2

Since $2 \neq - 8$ , there are no solutions.

No solution

Step 7.5

Solve $x + 2 = - (x - 8)$ for $x$ .

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Step 7.5.1

Simplify $- (x - 8)$ .

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Step 7.5.1.1

Rewrite.

$x + 2 = 0 + 0 - (x - 8)$

Step 7.5.1.2

Simplify by adding zeros.

$x + 2 = - (x - 8)$

Step 7.5.1.3

Apply the distributive property.

$x + 2 = - x - - 8$

Step 7.5.1.4

Multiply $- 1$ by $- 8$ .

$x + 2 = - x + 8$

Step 7.5.2

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

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Step 7.5.2.1

Add $x$ to both sides of the equation.

$x + 2 + x = 8$

Step 7.5.2.2

Add $x$ and $x$ .

$2 x + 2 = 8$

Step 7.5.3

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

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Step 7.5.3.1

Subtract $2$ from both sides of the equation.

$2 x = 8 - 2$

Step 7.5.3.2

Subtract $2$ from $8$ .

$2 x = 6$

Step 7.5.4

Divide each term in $2 x = 6$ by $2$ and simplify.

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Step 7.5.4.1

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

Step 7.5.4.2

Simplify the left side.

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Step 7.5.4.2.1

Cancel the common factor of $2$ .

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Step 7.5.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2}$

Step 7.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Step 7.5.4.3

Simplify the right side.

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Step 7.5.4.3.1

Divide $6$ by $2$ .

$x = 3$

Step 7.6

Consolidate the solutions.

$x = 3$

Step 8

Consolidate the solutions.

$x = - 2, 3$

Step 9

The solution consists of all of the true intervals.

$x \leq - 2$

Step 10

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

$x = - 2$

Step 11