Algebra Examples

$| x + 2 | + | x - 5 | + | x - 1 | = 12$

Step 1

Move all terms not containing $| x - 1 |$ to the right side of the equation.

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Step 1.1

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

$| x - 5 | + | x - 1 | = 12 - | x + 2 |$

Step 1.2

Subtract $| x - 5 |$ from both sides of the equation.

$| x - 1 | = 12 - | x + 2 | - | x - 5 |$

Step 2

Solve for $| x - 5 |$ .

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Step 2.1

Rewrite the equation as $12 - | x + 2 | - | x - 5 | = | x - 1 |$ .

$12 - | x + 2 | - | x - 5 | = | x - 1 |$

Step 2.2

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

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Step 2.2.1

Subtract $12$ from both sides of the equation.

$- | x + 2 | - | x - 5 | = | x - 1 | - 12$

Step 2.2.2

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

$- | x - 5 | = | x - 1 | - 12 + | x + 2 |$

Step 2.3

Divide each term in $- | x - 5 | = | x - 1 | - 12 + | x + 2 |$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- | x - 5 | = | x - 1 | - 12 + | x + 2 |$ by $- 1$ .

$\frac{- | x - 5 |}{- 1} = \frac{| x - 1 |}{- 1} + \frac{- 12}{- 1} + \frac{| x + 2 |}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{| x - 5 |}{1} = \frac{| x - 1 |}{- 1} + \frac{- 12}{- 1} + \frac{| x + 2 |}{- 1}$

Step 2.3.2.2

Divide $| x - 5 |$ by $1$ .

$| x - 5 | = \frac{| x - 1 |}{- 1} + \frac{- 12}{- 1} + \frac{| x + 2 |}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Move the negative one from the denominator of $\frac{| x - 1 |}{- 1}$ .

$| x - 5 | = - 1 \cdot | x - 1 | + \frac{- 12}{- 1} + \frac{| x + 2 |}{- 1}$

Step 2.3.3.1.2

Rewrite $- 1 \cdot | x - 1 |$ as $- | x - 1 |$ .

$| x - 5 | = - | x - 1 | + \frac{- 12}{- 1} + \frac{| x + 2 |}{- 1}$

Step 2.3.3.1.3

Divide $- 12$ by $- 1$ .

$| x - 5 | = - | x - 1 | + 12 + \frac{| x + 2 |}{- 1}$

Step 2.3.3.1.4

Move the negative one from the denominator of $\frac{| x + 2 |}{- 1}$ .

$| x - 5 | = - | x - 1 | + 12 - 1 \cdot | x + 2 |$

Step 2.3.3.1.5

Rewrite $- 1 \cdot | x + 2 |$ as $- | x + 2 |$ .

$| x - 5 | = - | x - 1 | + 12 - | x + 2 |$

Step 3

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

$x - 5 = \pm (- | x - 1 | + 12 - | x + 2 |)$

Step 4

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

$x - 5 = - | x - 1 | + 12 - | x + 2 |$

$x - 5 = - (- | x - 1 | + 12 - | x + 2 |)$

Step 5

Solve $x - 5 = - | x - 1 | + 12 - | x + 2 |$ for $x$ .

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Step 5.1

Solve for $| x + 2 |$ .

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Step 5.1.1

Rewrite the equation as $- | x - 1 | + 12 - | x + 2 | = x - 5$ .

$- | x - 1 | + 12 - | x + 2 | = x - 5$

Step 5.1.2

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

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Step 5.1.2.1

Add $| x - 1 |$ to both sides of the equation.

$12 - | x + 2 | = x - 5 + | x - 1 |$

Step 5.1.2.2

Subtract $12$ from both sides of the equation.

$- | x + 2 | = x - 5 + | x - 1 | - 12$

Step 5.1.2.3

Subtract $12$ from $- 5$ .

$- | x + 2 | = x + | x - 1 | - 17$

Step 5.1.3

Divide each term in $- | x + 2 | = x + | x - 1 | - 17$ by $- 1$ and simplify.

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Step 5.1.3.1

Divide each term in $- | x + 2 | = x + | x - 1 | - 17$ by $- 1$ .

$\frac{- | x + 2 |}{- 1} = \frac{x}{- 1} + \frac{| x - 1 |}{- 1} + \frac{- 17}{- 1}$

Step 5.1.3.2

Simplify the left side.

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Step 5.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{| x + 2 |}{1} = \frac{x}{- 1} + \frac{| x - 1 |}{- 1} + \frac{- 17}{- 1}$

Step 5.1.3.2.2

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

$| x + 2 | = \frac{x}{- 1} + \frac{| x - 1 |}{- 1} + \frac{- 17}{- 1}$

Step 5.1.3.3

Simplify the right side.

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Step 5.1.3.3.1

Simplify each term.

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Step 5.1.3.3.1.1

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$| x + 2 | = - 1 \cdot x + \frac{| x - 1 |}{- 1} + \frac{- 17}{- 1}$

Step 5.1.3.3.1.2

Rewrite $- 1 \cdot x$ as $- x$ .

$| x + 2 | = - x + \frac{| x - 1 |}{- 1} + \frac{- 17}{- 1}$

Step 5.1.3.3.1.3

Move the negative one from the denominator of $\frac{| x - 1 |}{- 1}$ .

$| x + 2 | = - x - 1 \cdot | x - 1 | + \frac{- 17}{- 1}$

Step 5.1.3.3.1.4

Rewrite $- 1 \cdot | x - 1 |$ as $- | x - 1 |$ .

$| x + 2 | = - x - | x - 1 | + \frac{- 17}{- 1}$

Step 5.1.3.3.1.5

Divide $- 17$ by $- 1$ .

$| x + 2 | = - x - | x - 1 | + 17$

Step 5.2

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

$x + 2 = \pm (- x - | x - 1 | + 17)$

Step 5.3

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

$x + 2 = - x - | x - 1 | + 17$

$x + 2 = - (- x - | x - 1 | + 17)$

Step 5.4

Solve $x + 2 = - x - | x - 1 | + 17$ for $x$ .

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Step 5.4.1

Solve for $| x - 1 |$ .

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Step 5.4.1.1

Rewrite the equation as $- x - | x - 1 | + 17 = x + 2$ .

$- x - | x - 1 | + 17 = x + 2$

Step 5.4.1.2

Move all terms not containing $| x - 1 |$ to the right side of the equation.

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Step 5.4.1.2.1

Add $x$ to both sides of the equation.

$- | x - 1 | + 17 = x + 2 + x$

Step 5.4.1.2.2

Subtract $17$ from both sides of the equation.

$- | x - 1 | = x + 2 + x - 17$

Step 5.4.1.2.3

Add $x$ and $x$ .

$- | x - 1 | = 2 x + 2 - 17$

Step 5.4.1.2.4

Subtract $17$ from $2$ .

$- | x - 1 | = 2 x - 15$

Step 5.4.1.3

Divide each term in $- | x - 1 | = 2 x - 15$ by $- 1$ and simplify.

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Step 5.4.1.3.1

Divide each term in $- | x - 1 | = 2 x - 15$ by $- 1$ .

$\frac{- | x - 1 |}{- 1} = \frac{2 x}{- 1} + \frac{- 15}{- 1}$

Step 5.4.1.3.2

Simplify the left side.

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Step 5.4.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{| x - 1 |}{1} = \frac{2 x}{- 1} + \frac{- 15}{- 1}$

Step 5.4.1.3.2.2

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

$| x - 1 | = \frac{2 x}{- 1} + \frac{- 15}{- 1}$

Step 5.4.1.3.3

Simplify the right side.

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Step 5.4.1.3.3.1

Simplify each term.

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Step 5.4.1.3.3.1.1

Move the negative one from the denominator of $\frac{2 x}{- 1}$ .

$| x - 1 | = - 1 \cdot (2 x) + \frac{- 15}{- 1}$

Step 5.4.1.3.3.1.2

Rewrite $- 1 \cdot (2 x)$ as $- (2 x)$ .

$| x - 1 | = - (2 x) + \frac{- 15}{- 1}$

Step 5.4.1.3.3.1.3

Multiply $2$ by $- 1$ .

$| x - 1 | = - 2 x + \frac{- 15}{- 1}$

Step 5.4.1.3.3.1.4

Divide $- 15$ by $- 1$ .

$| x - 1 | = - 2 x + 15$

Step 5.4.2

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

$x - 1 = \pm (- 2 x + 15)$

Step 5.4.3

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

$x - 1 = - 2 x + 15$

$x - 1 = - (- 2 x + 15)$

Step 5.4.4

Solve $x - 1 = - 2 x + 15$ for $x$ .

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Step 5.4.4.1

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

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Step 5.4.4.1.1

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

$x - 1 + 2 x = 15$

Step 5.4.4.1.2

Add $x$ and $2 x$ .

$3 x - 1 = 15$

Step 5.4.4.2

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

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Step 5.4.4.2.1

Add $1$ to both sides of the equation.

$3 x = 15 + 1$

Step 5.4.4.2.2

Add $15$ and $1$ .

$3 x = 16$

Step 5.4.4.3

Divide each term in $3 x = 16$ by $3$ and simplify.

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Step 5.4.4.3.1

Divide each term in $3 x = 16$ by $3$ .

$\frac{3 x}{3} = \frac{16}{3}$

Step 5.4.4.3.2

Simplify the left side.

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Step 5.4.4.3.2.1

Cancel the common factor of $3$ .

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Step 5.4.4.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{16}{3}$

Step 5.4.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{3}$

Step 5.4.5

Solve $x - 1 = - (- 2 x + 15)$ for $x$ .

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Step 5.4.5.1

Simplify $- (- 2 x + 15)$ .

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Step 5.4.5.1.1

Rewrite.

$x - 1 = 0 + 0 - (- 2 x + 15)$

Step 5.4.5.1.2

Simplify by adding zeros.

$x - 1 = - (- 2 x + 15)$

Step 5.4.5.1.3

Apply the distributive property.

$x - 1 = - (- 2 x) - 1 \cdot 15$

Step 5.4.5.1.4

Multiply.

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Step 5.4.5.1.4.1

Multiply $- 2$ by $- 1$ .

$x - 1 = 2 x - 1 \cdot 15$

Step 5.4.5.1.4.2

Multiply $- 1$ by $15$ .

$x - 1 = 2 x - 15$

Step 5.4.5.2

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

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Step 5.4.5.2.1

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

$x - 1 - 2 x = - 15$

Step 5.4.5.2.2

Subtract $2 x$ from $x$ .

$- x - 1 = - 15$

Step 5.4.5.3

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

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Step 5.4.5.3.1

Add $1$ to both sides of the equation.

$- x = - 15 + 1$

Step 5.4.5.3.2

Add $- 15$ and $1$ .

$- x = - 14$

Step 5.4.5.4

Divide each term in $- x = - 14$ by $- 1$ and simplify.

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Step 5.4.5.4.1

Divide each term in $- x = - 14$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 14}{- 1}$

Step 5.4.5.4.2

Simplify the left side.

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Step 5.4.5.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 14}{- 1}$

Step 5.4.5.4.2.2

Divide $x$ by $1$ .

$x = \frac{- 14}{- 1}$

Step 5.4.5.4.3

Simplify the right side.

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Step 5.4.5.4.3.1

Divide $- 14$ by $- 1$ .

$x = 14$

Step 5.4.6

Consolidate the solutions.

$x = \frac{16}{3}, 14$

Step 5.5

Solve $x + 2 = - (- x - | x - 1 | + 17)$ for $x$ .

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Step 5.5.1

Solve for $| x - 1 |$ .

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Step 5.5.1.1

Rewrite the equation as $- (- x - | x - 1 | + 17) = x + 2$ .

$- (- x - | x - 1 | + 17) = x + 2$

Step 5.5.1.2

Simplify $- (- x - | x - 1 | + 17)$ .

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Step 5.5.1.2.1

Apply the distributive property.

$- - x - - | x - 1 | - 1 \cdot 17 = x + 2$

Step 5.5.1.2.2

Simplify.

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Step 5.5.1.2.2.1

Multiply $- - x$ .

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Step 5.5.1.2.2.1.1

Multiply $- 1$ by $- 1$ .

$1 x - - | x - 1 | - 1 \cdot 17 = x + 2$

Step 5.5.1.2.2.1.2

Multiply $x$ by $1$ .

$x - - | x - 1 | - 1 \cdot 17 = x + 2$

Step 5.5.1.2.2.2

Multiply $- - | x - 1 |$ .

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Step 5.5.1.2.2.2.1

Multiply $- 1$ by $- 1$ .

$x + 1 | x - 1 | - 1 \cdot 17 = x + 2$

Step 5.5.1.2.2.2.2

Multiply $| x - 1 |$ by $1$ .

$x + | x - 1 | - 1 \cdot 17 = x + 2$

Step 5.5.1.2.2.3

Multiply $- 1$ by $17$ .

$x + | x - 1 | - 17 = x + 2$

Step 5.5.1.3

Move all terms not containing $| x - 1 |$ to the right side of the equation.

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Step 5.5.1.3.1

Subtract $x$ from both sides of the equation.

$| x - 1 | - 17 = x + 2 - x$

Step 5.5.1.3.2

Add $17$ to both sides of the equation.

$| x - 1 | = x + 2 - x + 17$

Step 5.5.1.3.3

Combine the opposite terms in $x + 2 - x + 17$ .

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Step 5.5.1.3.3.1

Subtract $x$ from $x$ .

$| x - 1 | = 0 + 2 + 17$

Step 5.5.1.3.3.2

Add $0$ and $2$ .

$| x - 1 | = 2 + 17$

Step 5.5.1.3.4

Add $2$ and $17$ .

$| x - 1 | = 19$

Step 5.5.2

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

$x - 1 = \pm (19)$

Step 5.5.3

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

$x - 1 = 19$

$x - 1 = - (19)$

Step 5.5.4

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

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Step 5.5.4.1

Add $1$ to both sides of the equation.

$x = 19 + 1$

Step 5.5.4.2

Add $19$ and $1$ .

$x = 20$

Step 5.5.5

Solve $x - 1 = - (19)$ for $x$ .

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Step 5.5.5.1

Multiply $- 1$ by $19$ .

$x - 1 = - 19$

Step 5.5.5.2

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

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Step 5.5.5.2.1

Add $1$ to both sides of the equation.

$x = - 19 + 1$

Step 5.5.5.2.2

Add $- 19$ and $1$ .

$x = - 18$

Step 5.5.6

Consolidate the solutions.

$x = 20, - 18$

Step 5.6

Consolidate the solutions.

$x = \frac{16}{3}, 14, 20, - 18$

Step 6

Solve $x - 5 = - (- | x - 1 | + 12 - | x + 2 |)$ 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 $- (- | x - 1 | + 12 - | x + 2 |) = x - 5$ .

$- (- | x - 1 | + 12 - | x + 2 |) = x - 5$

Step 6.1.2

Simplify $- (- | x - 1 | + 12 - | x + 2 |)$ .

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Step 6.1.2.1

Apply the distributive property.

$- - | x - 1 | - 1 \cdot 12 - - | x + 2 | = x - 5$

Step 6.1.2.2

Simplify.

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Step 6.1.2.2.1

Multiply $- - | x - 1 |$ .

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Step 6.1.2.2.1.1

Multiply $- 1$ by $- 1$ .

$1 | x - 1 | - 1 \cdot 12 - - | x + 2 | = x - 5$

Step 6.1.2.2.1.2

Multiply $| x - 1 |$ by $1$ .

$| x - 1 | - 1 \cdot 12 - - | x + 2 | = x - 5$

Step 6.1.2.2.2

Multiply $- 1$ by $12$ .

$| x - 1 | - 12 - - | x + 2 | = x - 5$

Step 6.1.2.2.3

Multiply $- - | x + 2 |$ .

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Step 6.1.2.2.3.1

Multiply $- 1$ by $- 1$ .

$| x - 1 | - 12 + 1 | x + 2 | = x - 5$

Step 6.1.2.2.3.2

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

$| x - 1 | - 12 + | x + 2 | = x - 5$

Step 6.1.3

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

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Step 6.1.3.1

Subtract $| x - 1 |$ from both sides of the equation.

$- 12 + | x + 2 | = x - 5 - | x - 1 |$

Step 6.1.3.2

Add $12$ to both sides of the equation.

$| x + 2 | = x - 5 - | x - 1 | + 12$

Step 6.1.3.3

Add $- 5$ and $12$ .

$| x + 2 | = x - | x - 1 | + 7$

Step 6.2

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

$x + 2 = \pm (x - | x - 1 | + 7)$

Step 6.3

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

$x + 2 = x - | x - 1 | + 7$

$x + 2 = - (x - | x - 1 | + 7)$

Step 6.4

Solve $x + 2 = x - | x - 1 | + 7$ for $x$ .

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Step 6.4.1

Solve for $| x - 1 |$ .

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Step 6.4.1.1

Rewrite the equation as $x - | x - 1 | + 7 = x + 2$ .

$x - | x - 1 | + 7 = x + 2$

Step 6.4.1.2

Move all terms not containing $| x - 1 |$ to the right side of the equation.

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Step 6.4.1.2.1

Subtract $x$ from both sides of the equation.

$- | x - 1 | + 7 = x + 2 - x$

Step 6.4.1.2.2

Subtract $7$ from both sides of the equation.

$- | x - 1 | = x + 2 - x - 7$

Step 6.4.1.2.3

Combine the opposite terms in $x + 2 - x - 7$ .

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Step 6.4.1.2.3.1

Subtract $x$ from $x$ .

$- | x - 1 | = 0 + 2 - 7$

Step 6.4.1.2.3.2

Add $0$ and $2$ .

$- | x - 1 | = 2 - 7$

Step 6.4.1.2.4

Subtract $7$ from $2$ .

$- | x - 1 | = - 5$

Step 6.4.1.3

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

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Step 6.4.1.3.1

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

$\frac{- | x - 1 |}{- 1} = \frac{- 5}{- 1}$

Step 6.4.1.3.2

Simplify the left side.

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Step 6.4.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{| x - 1 |}{1} = \frac{- 5}{- 1}$

Step 6.4.1.3.2.2

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

$| x - 1 | = \frac{- 5}{- 1}$

Step 6.4.1.3.3

Simplify the right side.

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Step 6.4.1.3.3.1

Divide $- 5$ by $- 1$ .

$| x - 1 | = 5$

Step 6.4.2

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

$x - 1 = \pm (5)$

Step 6.4.3

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

$x - 1 = 5$

$x - 1 = - (5)$

Step 6.4.4

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

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Step 6.4.4.1

Add $1$ to both sides of the equation.

$x = 5 + 1$

Step 6.4.4.2

Add $5$ and $1$ .

$x = 6$

Step 6.4.5

Solve $x - 1 = - (5)$ for $x$ .

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Step 6.4.5.1

Multiply $- 1$ by $5$ .

$x - 1 = - 5$

Step 6.4.5.2

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

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Step 6.4.5.2.1

Add $1$ to both sides of the equation.

$x = - 5 + 1$

Step 6.4.5.2.2

Add $- 5$ and $1$ .

$x = - 4$

Step 6.4.6

Consolidate the solutions.

$x = 6, - 4$

Step 6.5

Solve $x + 2 = - (x - | x - 1 | + 7)$ for $x$ .

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Step 6.5.1

Solve for $| x - 1 |$ .

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Step 6.5.1.1

Rewrite the equation as $- (x - | x - 1 | + 7) = x + 2$ .

$- (x - | x - 1 | + 7) = x + 2$

Step 6.5.1.2

Simplify $- (x - | x - 1 | + 7)$ .

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Step 6.5.1.2.1

Apply the distributive property.

$- x - - | x - 1 | - 1 \cdot 7 = x + 2$

Step 6.5.1.2.2

Simplify.

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Step 6.5.1.2.2.1

Multiply $- - | x - 1 |$ .

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Step 6.5.1.2.2.1.1

Multiply $- 1$ by $- 1$ .

$- x + 1 | x - 1 | - 1 \cdot 7 = x + 2$

Step 6.5.1.2.2.1.2

Multiply $| x - 1 |$ by $1$ .

$- x + | x - 1 | - 1 \cdot 7 = x + 2$

Step 6.5.1.2.2.2

Multiply $- 1$ by $7$ .

$- x + | x - 1 | - 7 = x + 2$

Step 6.5.1.3

Move all terms not containing $| x - 1 |$ to the right side of the equation.

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Step 6.5.1.3.1

Add $x$ to both sides of the equation.

$| x - 1 | - 7 = x + 2 + x$

Step 6.5.1.3.2

Add $7$ to both sides of the equation.

$| x - 1 | = x + 2 + x + 7$

Step 6.5.1.3.3

Add $x$ and $x$ .

$| x - 1 | = 2 x + 2 + 7$

Step 6.5.1.3.4

Add $2$ and $7$ .

$| x - 1 | = 2 x + 9$

Step 6.5.2

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

$x - 1 = \pm (2 x + 9)$

Step 6.5.3

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

$x - 1 = 2 x + 9$

$x - 1 = - (2 x + 9)$

Step 6.5.4

Solve $x - 1 = 2 x + 9$ for $x$ .

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Step 6.5.4.1

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

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Step 6.5.4.1.1

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

$x - 1 - 2 x = 9$

Step 6.5.4.1.2

Subtract $2 x$ from $x$ .

$- x - 1 = 9$

Step 6.5.4.2

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

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Step 6.5.4.2.1

Add $1$ to both sides of the equation.

$- x = 9 + 1$

Step 6.5.4.2.2

Add $9$ and $1$ .

$- x = 10$

Step 6.5.4.3

Divide each term in $- x = 10$ by $- 1$ and simplify.

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Step 6.5.4.3.1

Divide each term in $- x = 10$ by $- 1$ .

$\frac{- x}{- 1} = \frac{10}{- 1}$

Step 6.5.4.3.2

Simplify the left side.

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Step 6.5.4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{10}{- 1}$

Step 6.5.4.3.2.2

Divide $x$ by $1$ .

$x = \frac{10}{- 1}$

Step 6.5.4.3.3

Simplify the right side.

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Step 6.5.4.3.3.1

Divide $10$ by $- 1$ .

$x = - 10$

Step 6.5.5

Solve $x - 1 = - (2 x + 9)$ for $x$ .

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Step 6.5.5.1

Simplify $- (2 x + 9)$ .

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Step 6.5.5.1.1

Rewrite.

$x - 1 = 0 + 0 - (2 x + 9)$

Step 6.5.5.1.2

Simplify by adding zeros.

$x - 1 = - (2 x + 9)$

Step 6.5.5.1.3

Apply the distributive property.

$x - 1 = - (2 x) - 1 \cdot 9$

Step 6.5.5.1.4

Multiply.

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Step 6.5.5.1.4.1

Multiply $2$ by $- 1$ .

$x - 1 = - 2 x - 1 \cdot 9$

Step 6.5.5.1.4.2

Multiply $- 1$ by $9$ .

$x - 1 = - 2 x - 9$

Step 6.5.5.2

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

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Step 6.5.5.2.1

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

$x - 1 + 2 x = - 9$

Step 6.5.5.2.2

Add $x$ and $2 x$ .

$3 x - 1 = - 9$

Step 6.5.5.3

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

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Step 6.5.5.3.1

Add $1$ to both sides of the equation.

$3 x = - 9 + 1$

Step 6.5.5.3.2

Add $- 9$ and $1$ .

$3 x = - 8$

Step 6.5.5.4

Divide each term in $3 x = - 8$ by $3$ and simplify.

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Step 6.5.5.4.1

Divide each term in $3 x = - 8$ by $3$ .

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

Step 6.5.5.4.2

Simplify the left side.

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Step 6.5.5.4.2.1

Cancel the common factor of $3$ .

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Step 6.5.5.4.2.1.1

Cancel the common factor.

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

Step 6.5.5.4.2.1.2

Divide $x$ by $1$ .

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

Step 6.5.5.4.3

Simplify the right side.

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Step 6.5.5.4.3.1

Move the negative in front of the fraction.

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

Step 6.5.6

Consolidate the solutions.

$x = - 10, - \frac{8}{3}$

Step 6.6

Consolidate the solutions.

$x = 6, - 4, - 10, - \frac{8}{3}$

Step 7

Consolidate the solutions.

$x = \frac{16}{3}, 14, 20, - 18, 6, - 4, - 10, - \frac{8}{3}$

Step 8

Exclude the solutions that do not make $| x + 2 | + | x - 5 | + | x - 1 | = 12$ true.

$x = \frac{16}{3}, - \frac{8}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{16}{3}, - \frac{8}{3}$

Decimal Form:

$x = 5.\overline{3}, - 2.\overline{6}$

Mixed Number Form:

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

Step 10