Find All Complex Solutions |3-x|-|x+2|=5

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

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.

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Simplify each term.

Divide $5$ by $- 1$ .

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

Dividing two negative values results in a positive value.

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

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

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

Step 3

Solve for $| 3 - x |$ .

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

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

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 $x$ .

Solve for $| x + 2 |$ .

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

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

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

Subtract $5$ from both sides of the equation.

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

Subtract $5$ from $3$ .

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

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

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

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

$x + 2 = - x - 2$

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

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

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

Add $x$ to both sides of the equation.

$x + 2 + x = - 2$

Add $x$ and $x$ .

$2 x + 2 = - 2$

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

Subtract $2$ from both sides of the equation.

$2 x = - 2 - 2$

Subtract $2$ from $- 2$ .

$2 x = - 4$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 4$ by $2$ .

$x = - 2$

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

Simplify $- (- x - 2)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$x + 2 = 1 x - - 2$

Multiply $x$ by $1$ .

$x + 2 = x - - 2$

Multiply $- 1$ by $- 2$ .

$x + 2 = x + 2$

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

Subtract $x$ from both sides of the equation.

$x + 2 - x = 2$

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

Subtract $x$ from $x$ .

$0 + 2 = 2$

Add $0$ and $2$ .

$2 = 2$

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

Always true

Consolidate the solutions.

$x = - 2$

Step 7

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

Solve for $| x + 2 |$ .

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $5$ .

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

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

Add $5$ to both sides of the equation.

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

Add $3$ and $5$ .

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

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Divide $8$ by $- 1$ .

$| x + 2 | = x - 8$

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

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

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

$x + 2 = x - 8$

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

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

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

Subtract $x$ from both sides of the equation.

$x + 2 - x = - 8$

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

Subtract $x$ from $x$ .

$0 + 2 = - 8$

Add $0$ and $2$ .

$2 = - 8$

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

No solution

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

Simplify $- (x - 8)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

$x + 2 = - x - - 8$

Multiply $- 1$ by $- 8$ .

$x + 2 = - x + 8$

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

Add $x$ to both sides of the equation.

$x + 2 + x = 8$

Add $x$ and $x$ .

$2 x + 2 = 8$

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

Subtract $2$ from both sides of the equation.

$2 x = 8 - 2$

Subtract $2$ from $8$ .

$2 x = 6$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $6$ by $2$ .

$x = 3$

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

Find All Complex Solutions |3-x|-|x+2|=5

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