Precalculus Examples

$| x - 2 | - | 3 x - 8 | + | x - 7 | = - 2$

Step 1

Move all terms not containing $| x - 7 |$ to the right side of the equation.

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Step 1.1

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

$- | 3 x - 8 | + | x - 7 | = - 2 - | x - 2 |$

Step 1.2

Add $| 3 x - 8 |$ to both sides of the equation.

$| x - 7 | = - 2 - | x - 2 | + | 3 x - 8 |$

Step 2

Solve for $| 3 x - 8 |$ .

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Step 2.1

Rewrite the equation as $- 2 - | x - 2 | + | 3 x - 8 | = | x - 7 |$ .

$- 2 - | x - 2 | + | 3 x - 8 | = | x - 7 |$

Step 2.2

Move all terms not containing $| 3 x - 8 |$ to the right side of the equation.

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Step 2.2.1

Add $2$ to both sides of the equation.

$- | x - 2 | + | 3 x - 8 | = | x - 7 | + 2$

Step 2.2.2

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

$| 3 x - 8 | = | x - 7 | + 2 + | x - 2 |$

Step 3

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

$3 x - 8 = \pm (| x - 7 | + 2 + | x - 2 |)$

Step 4

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

$3 x - 8 = | x - 7 | + 2 + | x - 2 |$

$3 x - 8 = - (| x - 7 | + 2 + | x - 2 |)$

Step 5

Solve $3 x - 8 = | x - 7 | + 2 + | x - 2 |$ 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 $| x - 7 | + 2 + | x - 2 | = 3 x - 8$ .

$| x - 7 | + 2 + | x - 2 | = 3 x - 8$

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

Subtract $| x - 7 |$ from both sides of the equation.

$2 + | x - 2 | = 3 x - 8 - | x - 7 |$

Step 5.1.2.2

Subtract $2$ from both sides of the equation.

$| x - 2 | = 3 x - 8 - | x - 7 | - 2$

Step 5.1.2.3

Subtract $2$ from $- 8$ .

$| x - 2 | = 3 x - | x - 7 | - 10$

Step 5.2

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

$x - 2 = \pm (3 x - | x - 7 | - 10)$

Step 5.3

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

$x - 2 = 3 x - | x - 7 | - 10$

$x - 2 = - (3 x - | x - 7 | - 10)$

Step 5.4

Solve $x - 2 = 3 x - | x - 7 | - 10$ for $x$ .

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Step 5.4.1

Solve for $| x - 7 |$ .

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Step 5.4.1.1

Rewrite the equation as $3 x - | x - 7 | - 10 = x - 2$ .

$3 x - | x - 7 | - 10 = x - 2$

Step 5.4.1.2

Move all terms not containing $| x - 7 |$ to the right side of the equation.

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Step 5.4.1.2.1

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

$- | x - 7 | - 10 = x - 2 - 3 x$

Step 5.4.1.2.2

Add $10$ to both sides of the equation.

$- | x - 7 | = x - 2 - 3 x + 10$

Step 5.4.1.2.3

Subtract $3 x$ from $x$ .

$- | x - 7 | = - 2 x - 2 + 10$

Step 5.4.1.2.4

Add $- 2$ and $10$ .

$- | x - 7 | = - 2 x + 8$

Step 5.4.1.3

Divide each term in $- | x - 7 | = - 2 x + 8$ by $- 1$ and simplify.

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Step 5.4.1.3.1

Divide each term in $- | x - 7 | = - 2 x + 8$ by $- 1$ .

$\frac{- | x - 7 |}{- 1} = \frac{- 2 x}{- 1} + \frac{8}{- 1}$

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.

$\frac{| x - 7 |}{1} = \frac{- 2 x}{- 1} + \frac{8}{- 1}$

Step 5.4.1.3.2.2

Divide $| x - 7 |$ by $1$ .

$| x - 7 | = \frac{- 2 x}{- 1} + \frac{8}{- 1}$

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 $\frac{- 2 x}{- 1}$ .

$| x - 7 | = - 1 \cdot (- 2 x) + \frac{8}{- 1}$

Step 5.4.1.3.3.1.2

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

$| x - 7 | = - (- 2 x) + \frac{8}{- 1}$

Step 5.4.1.3.3.1.3

Multiply $- 2$ by $- 1$ .

$| x - 7 | = 2 x + \frac{8}{- 1}$

Step 5.4.1.3.3.1.4

Divide $8$ by $- 1$ .

$| x - 7 | = 2 x - 8$

Step 5.4.2

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

$x - 7 = \pm (2 x - 8)$

Step 5.4.3

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

$x - 7 = 2 x - 8$

$x - 7 = - (2 x - 8)$

Step 5.4.4

Solve $x - 7 = 2 x - 8$ 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

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

$x - 7 - 2 x = - 8$

Step 5.4.4.1.2

Subtract $2 x$ from $x$ .

$- x - 7 = - 8$

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 $7$ to both sides of the equation.

$- x = - 8 + 7$

Step 5.4.4.2.2

Add $- 8$ and $7$ .

$- x = - 1$

Step 5.4.4.3

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

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Step 5.4.4.3.1

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

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

Step 5.4.4.3.2

Simplify the left side.

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Step 5.4.4.3.2.1

Dividing two negative values results in a positive value.

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

Step 5.4.4.3.2.2

Divide $x$ by $1$ .

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

Step 5.4.4.3.3

Simplify the right side.

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Step 5.4.4.3.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 5.4.5

Solve $x - 7 = - (2 x - 8)$ for $x$ .

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Step 5.4.5.1

Simplify $- (2 x - 8)$ .

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Step 5.4.5.1.1

Rewrite.

$x - 7 = 0 + 0 - (2 x - 8)$

Step 5.4.5.1.2

Simplify by adding zeros.

$x - 7 = - (2 x - 8)$

Step 5.4.5.1.3

Apply the distributive property.

$x - 7 = - (2 x) - - 8$

Step 5.4.5.1.4

Multiply.

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Step 5.4.5.1.4.1

Multiply $2$ by $- 1$ .

$x - 7 = - 2 x - - 8$

Step 5.4.5.1.4.2

Multiply $- 1$ by $- 8$ .

$x - 7 = - 2 x + 8$

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

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

$x - 7 + 2 x = 8$

Step 5.4.5.2.2

Add $x$ and $2 x$ .

$3 x - 7 = 8$

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 $7$ to both sides of the equation.

$3 x = 8 + 7$

Step 5.4.5.3.2

Add $8$ and $7$ .

$3 x = 15$

Step 5.4.5.4

Divide each term in $3 x = 15$ by $3$ and simplify.

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Step 5.4.5.4.1

Divide each term in $3 x = 15$ by $3$ .

$\frac{3 x}{3} = \frac{15}{3}$

Step 5.4.5.4.2

Simplify the left side.

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Step 5.4.5.4.2.1

Cancel the common factor of $3$ .

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Step 5.4.5.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{15}{3}$

Step 5.4.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{3}$

Step 5.4.5.4.3

Simplify the right side.

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Step 5.4.5.4.3.1

Divide $15$ by $3$ .

$x = 5$

Step 5.4.6

Consolidate the solutions.

$x = 1, 5$

Step 5.5

Solve $x - 2 = - (3 x - | x - 7 | - 10)$ for $x$ .

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Step 5.5.1

Solve for $| x - 7 |$ .

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Step 5.5.1.1

Rewrite the equation as $- (3 x - | x - 7 | - 10) = x - 2$ .

$- (3 x - | x - 7 | - 10) = x - 2$

Step 5.5.1.2

Simplify $- (3 x - | x - 7 | - 10)$ .

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Step 5.5.1.2.1

Apply the distributive property.

$- (3 x) - - | x - 7 | - - 10 = x - 2$

Step 5.5.1.2.2

Simplify.

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Step 5.5.1.2.2.1

Multiply $3$ by $- 1$ .

$- 3 x - - | x - 7 | - - 10 = x - 2$

Step 5.5.1.2.2.2

Multiply $- - | x - 7 |$ .

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Step 5.5.1.2.2.2.1

Multiply $- 1$ by $- 1$ .

$- 3 x + 1 | x - 7 | - - 10 = x - 2$

Step 5.5.1.2.2.2.2

Multiply $| x - 7 |$ by $1$ .

$- 3 x + | x - 7 | - - 10 = x - 2$

Step 5.5.1.2.2.3

Multiply $- 1$ by $- 10$ .

$- 3 x + | x - 7 | + 10 = x - 2$

Step 5.5.1.3

Move all terms not containing $| x - 7 |$ to the right side of the equation.

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Step 5.5.1.3.1

Add $3 x$ to both sides of the equation.

$| x - 7 | + 10 = x - 2 + 3 x$

Step 5.5.1.3.2

Subtract $10$ from both sides of the equation.

$| x - 7 | = x - 2 + 3 x - 10$

Step 5.5.1.3.3

Add $x$ and $3 x$ .

$| x - 7 | = 4 x - 2 - 10$

Step 5.5.1.3.4

Subtract $10$ from $- 2$ .

$| x - 7 | = 4 x - 12$

Step 5.5.2

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

$x - 7 = \pm (4 x - 12)$

Step 5.5.3

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

$x - 7 = 4 x - 12$

$x - 7 = - (4 x - 12)$

Step 5.5.4

Solve $x - 7 = 4 x - 12$ for $x$ .

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Step 5.5.4.1

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

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Step 5.5.4.1.1

Subtract $4 x$ from both sides of the equation.

$x - 7 - 4 x = - 12$

Step 5.5.4.1.2

Subtract $4 x$ from $x$ .

$- 3 x - 7 = - 12$

Step 5.5.4.2

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

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Step 5.5.4.2.1

Add $7$ to both sides of the equation.

$- 3 x = - 12 + 7$

Step 5.5.4.2.2

Add $- 12$ and $7$ .

$- 3 x = - 5$

Step 5.5.4.3

Divide each term in $- 3 x = - 5$ by $- 3$ and simplify.

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Step 5.5.4.3.1

Divide each term in $- 3 x = - 5$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 5}{- 3}$

Step 5.5.4.3.2

Simplify the left side.

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Step 5.5.4.3.2.1

Cancel the common factor of $- 3$ .

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Step 5.5.4.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 5}{- 3}$

Step 5.5.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 3}$

Step 5.5.4.3.3

Simplify the right side.

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Step 5.5.4.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{5}{3}$

Step 5.5.5

Solve $x - 7 = - (4 x - 12)$ for $x$ .

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Step 5.5.5.1

Simplify $- (4 x - 12)$ .

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Step 5.5.5.1.1

Rewrite.

$x - 7 = 0 + 0 - (4 x - 12)$

Step 5.5.5.1.2

Simplify by adding zeros.

$x - 7 = - (4 x - 12)$

Step 5.5.5.1.3

Apply the distributive property.

$x - 7 = - (4 x) - - 12$

Step 5.5.5.1.4

Multiply.

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Step 5.5.5.1.4.1

Multiply $4$ by $- 1$ .

$x - 7 = - 4 x - - 12$

Step 5.5.5.1.4.2

Multiply $- 1$ by $- 12$ .

$x - 7 = - 4 x + 12$

Step 5.5.5.2

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

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Step 5.5.5.2.1

Add $4 x$ to both sides of the equation.

$x - 7 + 4 x = 12$

Step 5.5.5.2.2

Add $x$ and $4 x$ .

$5 x - 7 = 12$

Step 5.5.5.3

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

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Step 5.5.5.3.1

Add $7$ to both sides of the equation.

$5 x = 12 + 7$

Step 5.5.5.3.2

Add $12$ and $7$ .

$5 x = 19$

Step 5.5.5.4

Divide each term in $5 x = 19$ by $5$ and simplify.

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Step 5.5.5.4.1

Divide each term in $5 x = 19$ by $5$ .

$\frac{5 x}{5} = \frac{19}{5}$

Step 5.5.5.4.2

Simplify the left side.

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Step 5.5.5.4.2.1

Cancel the common factor of $5$ .

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Step 5.5.5.4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{19}{5}$

Step 5.5.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{19}{5}$

Step 5.5.6

Consolidate the solutions.

$x = \frac{5}{3}, \frac{19}{5}$

Step 5.6

Consolidate the solutions.

$x = 1, 5, \frac{5}{3}, \frac{19}{5}$

Step 6

Solve $3 x - 8 = - (| x - 7 | + 2 + | x - 2 |)$ 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 $- (| x - 7 | + 2 + | x - 2 |) = 3 x - 8$ .

$- (| x - 7 | + 2 + | x - 2 |) = 3 x - 8$

Step 6.1.2

Simplify $- (| x - 7 | + 2 + | x - 2 |)$ .

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Step 6.1.2.1

Apply the distributive property.

$- | x - 7 | - 1 \cdot 2 - | x - 2 | = 3 x - 8$

Step 6.1.2.2

Multiply $- 1$ by $2$ .

$- | x - 7 | - 2 - | x - 2 | = 3 x - 8$

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

Add $| x - 7 |$ to both sides of the equation.

$- 2 - | x - 2 | = 3 x - 8 + | x - 7 |$

Step 6.1.3.2

Add $2$ to both sides of the equation.

$- | x - 2 | = 3 x - 8 + | x - 7 | + 2$

Step 6.1.3.3

Add $- 8$ and $2$ .

$- | x - 2 | = 3 x + | x - 7 | - 6$

Step 6.1.4

Divide each term in $- | x - 2 | = 3 x + | x - 7 | - 6$ by $- 1$ and simplify.

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Step 6.1.4.1

Divide each term in $- | x - 2 | = 3 x + | x - 7 | - 6$ by $- 1$ .

$\frac{- | x - 2 |}{- 1} = \frac{3 x}{- 1} + \frac{| x - 7 |}{- 1} + \frac{- 6}{- 1}$

Step 6.1.4.2

Simplify the left side.

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Step 6.1.4.2.1

Dividing two negative values results in a positive value.

$\frac{| x - 2 |}{1} = \frac{3 x}{- 1} + \frac{| x - 7 |}{- 1} + \frac{- 6}{- 1}$

Step 6.1.4.2.2

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

$| x - 2 | = \frac{3 x}{- 1} + \frac{| x - 7 |}{- 1} + \frac{- 6}{- 1}$

Step 6.1.4.3

Simplify the right side.

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Step 6.1.4.3.1

Simplify each term.

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Step 6.1.4.3.1.1

Move the negative one from the denominator of $\frac{3 x}{- 1}$ .

$| x - 2 | = - 1 \cdot (3 x) + \frac{| x - 7 |}{- 1} + \frac{- 6}{- 1}$

Step 6.1.4.3.1.2

Rewrite $- 1 \cdot (3 x)$ as $- (3 x)$ .

$| x - 2 | = - (3 x) + \frac{| x - 7 |}{- 1} + \frac{- 6}{- 1}$

Step 6.1.4.3.1.3

Multiply $3$ by $- 1$ .

$| x - 2 | = - 3 x + \frac{| x - 7 |}{- 1} + \frac{- 6}{- 1}$

Step 6.1.4.3.1.4

Move the negative one from the denominator of $\frac{| x - 7 |}{- 1}$ .

$| x - 2 | = - 3 x - 1 \cdot | x - 7 | + \frac{- 6}{- 1}$

Step 6.1.4.3.1.5

Rewrite $- 1 \cdot | x - 7 |$ as $- | x - 7 |$ .

$| x - 2 | = - 3 x - | x - 7 | + \frac{- 6}{- 1}$

Step 6.1.4.3.1.6

Divide $- 6$ by $- 1$ .

$| x - 2 | = - 3 x - | x - 7 | + 6$

Step 6.2

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

$x - 2 = \pm (- 3 x - | x - 7 | + 6)$

Step 6.3

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

$x - 2 = - 3 x - | x - 7 | + 6$

$x - 2 = - (- 3 x - | x - 7 | + 6)$

Step 6.4

Solve $x - 2 = - 3 x - | x - 7 | + 6$ for $x$ .

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Step 6.4.1

Solve for $| x - 7 |$ .

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Step 6.4.1.1

Rewrite the equation as $- 3 x - | x - 7 | + 6 = x - 2$ .

$- 3 x - | x - 7 | + 6 = x - 2$

Step 6.4.1.2

Move all terms not containing $| x - 7 |$ to the right side of the equation.

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Step 6.4.1.2.1

Add $3 x$ to both sides of the equation.

$- | x - 7 | + 6 = x - 2 + 3 x$

Step 6.4.1.2.2

Subtract $6$ from both sides of the equation.

$- | x - 7 | = x - 2 + 3 x - 6$

Step 6.4.1.2.3

Add $x$ and $3 x$ .

$- | x - 7 | = 4 x - 2 - 6$

Step 6.4.1.2.4

Subtract $6$ from $- 2$ .

$- | x - 7 | = 4 x - 8$

Step 6.4.1.3

Divide each term in $- | x - 7 | = 4 x - 8$ by $- 1$ and simplify.

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Step 6.4.1.3.1

Divide each term in $- | x - 7 | = 4 x - 8$ by $- 1$ .

$\frac{- | x - 7 |}{- 1} = \frac{4 x}{- 1} + \frac{- 8}{- 1}$

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.

$\frac{| x - 7 |}{1} = \frac{4 x}{- 1} + \frac{- 8}{- 1}$

Step 6.4.1.3.2.2

Divide $| x - 7 |$ by $1$ .

$| x - 7 | = \frac{4 x}{- 1} + \frac{- 8}{- 1}$

Step 6.4.1.3.3

Simplify the right side.

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Step 6.4.1.3.3.1

Simplify each term.

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Step 6.4.1.3.3.1.1

Move the negative one from the denominator of $\frac{4 x}{- 1}$ .

$| x - 7 | = - 1 \cdot (4 x) + \frac{- 8}{- 1}$

Step 6.4.1.3.3.1.2

Rewrite $- 1 \cdot (4 x)$ as $- (4 x)$ .

$| x - 7 | = - (4 x) + \frac{- 8}{- 1}$

Step 6.4.1.3.3.1.3

Multiply $4$ by $- 1$ .

$| x - 7 | = - 4 x + \frac{- 8}{- 1}$

Step 6.4.1.3.3.1.4

Divide $- 8$ by $- 1$ .

$| x - 7 | = - 4 x + 8$

Step 6.4.2

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

$x - 7 = \pm (- 4 x + 8)$

Step 6.4.3

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

$x - 7 = - 4 x + 8$

$x - 7 = - (- 4 x + 8)$

Step 6.4.4

Solve $x - 7 = - 4 x + 8$ for $x$ .

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Step 6.4.4.1

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

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Step 6.4.4.1.1

Add $4 x$ to both sides of the equation.

$x - 7 + 4 x = 8$

Step 6.4.4.1.2

Add $x$ and $4 x$ .

$5 x - 7 = 8$

Step 6.4.4.2

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

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Step 6.4.4.2.1

Add $7$ to both sides of the equation.

$5 x = 8 + 7$

Step 6.4.4.2.2

Add $8$ and $7$ .

$5 x = 15$

Step 6.4.4.3

Divide each term in $5 x = 15$ by $5$ and simplify.

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Step 6.4.4.3.1

Divide each term in $5 x = 15$ by $5$ .

$\frac{5 x}{5} = \frac{15}{5}$

Step 6.4.4.3.2

Simplify the left side.

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Step 6.4.4.3.2.1

Cancel the common factor of $5$ .

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Step 6.4.4.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{15}{5}$

Step 6.4.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{5}$

Step 6.4.4.3.3

Simplify the right side.

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Step 6.4.4.3.3.1

Divide $15$ by $5$ .

$x = 3$

Step 6.4.5

Solve $x - 7 = - (- 4 x + 8)$ for $x$ .

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Step 6.4.5.1

Simplify $- (- 4 x + 8)$ .

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Step 6.4.5.1.1

Rewrite.

$x - 7 = 0 + 0 - (- 4 x + 8)$

Step 6.4.5.1.2

Simplify by adding zeros.

$x - 7 = - (- 4 x + 8)$

Step 6.4.5.1.3

Apply the distributive property.

$x - 7 = - (- 4 x) - 1 \cdot 8$

Step 6.4.5.1.4

Multiply.

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Step 6.4.5.1.4.1

Multiply $- 4$ by $- 1$ .

$x - 7 = 4 x - 1 \cdot 8$

Step 6.4.5.1.4.2

Multiply $- 1$ by $8$ .

$x - 7 = 4 x - 8$

Step 6.4.5.2

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

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Step 6.4.5.2.1

Subtract $4 x$ from both sides of the equation.

$x - 7 - 4 x = - 8$

Step 6.4.5.2.2

Subtract $4 x$ from $x$ .

$- 3 x - 7 = - 8$

Step 6.4.5.3

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

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Step 6.4.5.3.1

Add $7$ to both sides of the equation.

$- 3 x = - 8 + 7$

Step 6.4.5.3.2

Add $- 8$ and $7$ .

$- 3 x = - 1$

Step 6.4.5.4

Divide each term in $- 3 x = - 1$ by $- 3$ and simplify.

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Step 6.4.5.4.1

Divide each term in $- 3 x = - 1$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 1}{- 3}$

Step 6.4.5.4.2

Simplify the left side.

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Step 6.4.5.4.2.1

Cancel the common factor of $- 3$ .

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Step 6.4.5.4.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 1}{- 3}$

Step 6.4.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 3}$

Step 6.4.5.4.3

Simplify the right side.

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Step 6.4.5.4.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{3}$

Step 6.4.6

Consolidate the solutions.

$x = 3, \frac{1}{3}$

Step 6.5

Solve $x - 2 = - (- 3 x - | x - 7 | + 6)$ for $x$ .

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Step 6.5.1

Solve for $| x - 7 |$ .

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Step 6.5.1.1

Rewrite the equation as $- (- 3 x - | x - 7 | + 6) = x - 2$ .

$- (- 3 x - | x - 7 | + 6) = x - 2$

Step 6.5.1.2

Simplify $- (- 3 x - | x - 7 | + 6)$ .

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Step 6.5.1.2.1

Apply the distributive property.

$- (- 3 x) - - | x - 7 | - 1 \cdot 6 = x - 2$

Step 6.5.1.2.2

Simplify.

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Step 6.5.1.2.2.1

Multiply $- 3$ by $- 1$ .

$3 x - - | x - 7 | - 1 \cdot 6 = x - 2$

Step 6.5.1.2.2.2

Multiply $- - | x - 7 |$ .

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Step 6.5.1.2.2.2.1

Multiply $- 1$ by $- 1$ .

$3 x + 1 | x - 7 | - 1 \cdot 6 = x - 2$

Step 6.5.1.2.2.2.2

Multiply $| x - 7 |$ by $1$ .

$3 x + | x - 7 | - 1 \cdot 6 = x - 2$

Step 6.5.1.2.2.3

Multiply $- 1$ by $6$ .

$3 x + | x - 7 | - 6 = x - 2$

Step 6.5.1.3

Move all terms not containing $| x - 7 |$ to the right side of the equation.

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Step 6.5.1.3.1

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

$| x - 7 | - 6 = x - 2 - 3 x$

Step 6.5.1.3.2

Add $6$ to both sides of the equation.

$| x - 7 | = x - 2 - 3 x + 6$

Step 6.5.1.3.3

Subtract $3 x$ from $x$ .

$| x - 7 | = - 2 x - 2 + 6$

Step 6.5.1.3.4

Add $- 2$ and $6$ .

$| x - 7 | = - 2 x + 4$

Step 6.5.2

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

$x - 7 = \pm (- 2 x + 4)$

Step 6.5.3

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

$x - 7 = - 2 x + 4$

$x - 7 = - (- 2 x + 4)$

Step 6.5.4

Solve $x - 7 = - 2 x + 4$ 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

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

$x - 7 + 2 x = 4$

Step 6.5.4.1.2

Add $x$ and $2 x$ .

$3 x - 7 = 4$

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 $7$ to both sides of the equation.

$3 x = 4 + 7$

Step 6.5.4.2.2

Add $4$ and $7$ .

$3 x = 11$

Step 6.5.4.3

Divide each term in $3 x = 11$ by $3$ and simplify.

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Step 6.5.4.3.1

Divide each term in $3 x = 11$ by $3$ .

$\frac{3 x}{3} = \frac{11}{3}$

Step 6.5.4.3.2

Simplify the left side.

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Step 6.5.4.3.2.1

Cancel the common factor of $3$ .

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Step 6.5.4.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{11}{3}$

Step 6.5.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{3}$

Step 6.5.5

Solve $x - 7 = - (- 2 x + 4)$ for $x$ .

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Step 6.5.5.1

Simplify $- (- 2 x + 4)$ .

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Step 6.5.5.1.1

Rewrite.

$x - 7 = 0 + 0 - (- 2 x + 4)$

Step 6.5.5.1.2

Simplify by adding zeros.

$x - 7 = - (- 2 x + 4)$

Step 6.5.5.1.3

Apply the distributive property.

$x - 7 = - (- 2 x) - 1 \cdot 4$

Step 6.5.5.1.4

Multiply.

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Step 6.5.5.1.4.1

Multiply $- 2$ by $- 1$ .

$x - 7 = 2 x - 1 \cdot 4$

Step 6.5.5.1.4.2

Multiply $- 1$ by $4$ .

$x - 7 = 2 x - 4$

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

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

$x - 7 - 2 x = - 4$

Step 6.5.5.2.2

Subtract $2 x$ from $x$ .

$- x - 7 = - 4$

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 $7$ to both sides of the equation.

$- x = - 4 + 7$

Step 6.5.5.3.2

Add $- 4$ and $7$ .

$- x = 3$

Step 6.5.5.4

Divide each term in $- x = 3$ by $- 1$ and simplify.

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Step 6.5.5.4.1

Divide each term in $- x = 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{3}{- 1}$

Step 6.5.5.4.2

Simplify the left side.

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Step 6.5.5.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{3}{- 1}$

Step 6.5.5.4.2.2

Divide $x$ by $1$ .

$x = \frac{3}{- 1}$

Step 6.5.5.4.3

Simplify the right side.

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Step 6.5.5.4.3.1

Divide $3$ by $- 1$ .

$x = - 3$

Step 6.5.6

Consolidate the solutions.

$x = \frac{11}{3}, - 3$

Step 6.6

Consolidate the solutions.

$x = 3, \frac{1}{3}, \frac{11}{3}, - 3$

Step 7

Consolidate the solutions.

$x = 1, 5, \frac{5}{3}, \frac{19}{5}, 3, \frac{1}{3}, \frac{11}{3}, - 3$

Step 8

Use each root to create test intervals.

$x < - 3$

$- 3 < x < \frac{1}{3}$

$\frac{1}{3} < x < 1$

$1 < x < \frac{5}{3}$

$\frac{5}{3} < x < 3$

$3 < x < \frac{11}{3}$

$\frac{11}{3} < x < \frac{19}{5}$

$\frac{19}{5} < x < 5$

$x > 5$

Step 9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 9.1

Test a value on the interval $x < - 3$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < - 3$ and see if this value makes the original inequality true.

$x = - 6$

Step 9.1.2

Replace $x$ with $- 6$ in the original inequality.

$| (- 6) - 2 | - | 3 (- 6) - 8 | + | (- 6) - 7 | = - 2$

Step 9.1.3

The left side $- 5$ does not equal to the right side $- 2$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $- 3 < x < \frac{1}{3}$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- 3 < x < \frac{1}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 9.2.2

Replace $x$ with $0$ in the original inequality.

$| (0) - 2 | - | 3 (0) - 8 | + | (0) - 7 | = - 2$

Step 9.2.3

The left side $1$ does not equal to the right side $- 2$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $\frac{1}{3} < x < 1$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $\frac{1}{3} < x < 1$ and see if this value makes the original inequality true.

$x = 0.67$

Step 9.3.2

Replace $x$ with $0.67$ in the original inequality.

$| (0.67) - 2 | - | 3 (0.67) - 8 | + | (0.67) - 7 | = - 2$

Step 9.3.3

The left side $1.67$ does not equal to the right side $- 2$ , which means that the given statement is false.

False

Step 9.4

Test a value on the interval $1 < x < \frac{5}{3}$ to see if it makes the inequality true.

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Step 9.4.1

Choose a value on the interval $1 < x < \frac{5}{3}$ and see if this value makes the original inequality true.

$x = 1.33$

Step 9.4.2

Replace $x$ with $1.33$ in the original inequality.

$| (1.33) - 2 | - | 3 (1.33) - 8 | + | (1.33) - 7 | = - 2$

Step 9.4.3

The left side $2.33$ does not equal to the right side $- 2$ , which means that the given statement is false.

False

Step 9.5

Test a value on the interval $\frac{5}{3} < x < 3$ to see if it makes the inequality true.

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Step 9.5.1

Choose a value on the interval $\frac{5}{3} < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Step 9.5.2

Replace $x$ with $2$ in the original inequality.

$| (2) - 2 | - | 3 (2) - 8 | + | (2) - 7 | = - 2$

Step 9.5.3

The left side $3$ does not equal to the right side $- 2$ , which means that the given statement is false.

False

Step 9.6

Test a value on the interval $3 < x < \frac{11}{3}$ to see if it makes the inequality true.

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Step 9.6.1

Choose a value on the interval $3 < x < \frac{11}{3}$ and see if this value makes the original inequality true.

$x = 3.33$

Step 9.6.2

Replace $x$ with $3.33$ in the original inequality.

$| (3.33) - 2 | - | 3 (3.33) - 8 | + | (3.33) - 7 | = - 2$

Step 9.6.3

The left side $3.01$ does not equal to the right side $- 2$ , which means that the given statement is false.

False

Step 9.7

Test a value on the interval $\frac{11}{3} < x < \frac{19}{5}$ to see if it makes the inequality true.

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Step 9.7.1

Choose a value on the interval $\frac{11}{3} < x < \frac{19}{5}$ and see if this value makes the original inequality true.

$x = 3.73$

Step 9.7.2

Replace $x$ with $3.73$ in the original inequality.

$| (3.73) - 2 | - | 3 (3.73) - 8 | + | (3.73) - 7 | = - 2$

Step 9.7.3

The left side $1.81$ does not equal to the right side $- 2$ , which means that the given statement is false.

False

Step 9.8

Test a value on the interval $\frac{19}{5} < x < 5$ to see if it makes the inequality true.

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Step 9.8.1

Choose a value on the interval $\frac{19}{5} < x < 5$ and see if this value makes the original inequality true.

$x = 4$

Step 9.8.2

Replace $x$ with $4$ in the original inequality.

$| (4) - 2 | - | 3 (4) - 8 | + | (4) - 7 | = - 2$

Step 9.8.3

The left side $1$ does not equal to the right side $- 2$ , which means that the given statement is false.

False

Step 9.9

Test a value on the interval $x > 5$ to see if it makes the inequality true.

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Step 9.9.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 9.9.2

Replace $x$ with $8$ in the original inequality.

$| (8) - 2 | - | 3 (8) - 8 | + | (8) - 7 | = - 2$

Step 9.9.3

The left side $- 9$ does not equal to the right side $- 2$ , which means that the given statement is false.

False

Step 9.10

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 3$ False

$- 3 < x < \frac{1}{3}$ False

$\frac{1}{3} < x < 1$ False

$1 < x < \frac{5}{3}$ False

$\frac{5}{3} < x < 3$ False

$3 < x < \frac{11}{3}$ False

$\frac{11}{3} < x < \frac{19}{5}$ False

$\frac{19}{5} < x < 5$ False

$x > 5$ False

$x < - 3$ False

$- 3 < x < \frac{1}{3}$ False

$\frac{1}{3} < x < 1$ False

$1 < x < \frac{5}{3}$ False

$\frac{5}{3} < x < 3$ False

$3 < x < \frac{11}{3}$ False

$\frac{11}{3} < x < \frac{19}{5}$ False

$\frac{19}{5} < x < 5$ False

$x > 5$ False

Step 10

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 11

Exclude the solutions that do not make $| x - 2 | - | 3 x - 8 | + | x - 7 | = - 2$ true.

$x = 5, - 3$

Step 12