Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2

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

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

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

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

Divide $- 3$ by $- 1$ .

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

Step 2.3.1.4

Dividing two negative values results in a positive value.

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

Step 2.3.1.5

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

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

Step 3

Solve for $| x + 7 |$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Add $x$ to both sides of the equation.

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

Step 3.2.2

Subtract $3$ from both sides of the equation.

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

Step 4

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

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

Step 5

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

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

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

Step 6

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

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

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 $x$ from both sides of the equation.

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

Step 6.1.2.2

Add $3$ to both sides of the equation.

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

Step 6.1.2.3

Combine the opposite terms in $x + 7 - x + 3$ .

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Step 6.1.2.3.1

Subtract $x$ from $x$ .

$| x - 2 | = 0 + 7 + 3$

Step 6.1.2.3.2

Add $0$ and $7$ .

$| x - 2 | = 7 + 3$

Step 6.1.2.4

Add $7$ and $3$ .

$| x - 2 | = 10$

Step 6.2

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

$x - 2 = \pm (10)$

Step 6.3

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

$x - 2 = 10$

$x - 2 = - (10)$

Step 6.4

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

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Step 6.4.1

Add $2$ to both sides of the equation.

$x = 10 + 2$

Step 6.4.2

Add $10$ and $2$ .

$x = 12$

Step 6.5

Solve $x - 2 = - (10)$ for $x$ .

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Step 6.5.1

Multiply $- 1$ by $10$ .

$x - 2 = - 10$

Step 6.5.2

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

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Step 6.5.2.1

Add $2$ to both sides of the equation.

$x = - 10 + 2$

Step 6.5.2.2

Add $- 10$ and $2$ .

$x = - 8$

Step 6.6

Consolidate the solutions.

$x = 12, - 8$

Step 7

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

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

Step 7.1.2

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

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Step 7.1.2.1

Apply the distributive property.

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

Step 7.1.2.2

Multiply $- 1$ by $- 3$ .

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

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 $x$ to both sides of the equation.

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

Step 7.1.3.2

Subtract $3$ from both sides of the equation.

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

Step 7.1.3.3

Add $x$ and $x$ .

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

Step 7.1.3.4

Subtract $3$ from $7$ .

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

Step 7.1.4

Divide each term in $- | x - 2 | = 2 x + 4$ by $- 1$ and simplify.

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Step 7.1.4.1

Divide each term in $- | x - 2 | = 2 x + 4$ by $- 1$ .

$\frac{- | x - 2 |}{- 1} = \frac{2 x}{- 1} + \frac{4}{- 1}$

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.

$\frac{| x - 2 |}{1} = \frac{2 x}{- 1} + \frac{4}{- 1}$

Step 7.1.4.2.2

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

$| x - 2 | = \frac{2 x}{- 1} + \frac{4}{- 1}$

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

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

$| x - 2 | = - 1 \cdot (2 x) + \frac{4}{- 1}$

Step 7.1.4.3.1.2

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

$| x - 2 | = - (2 x) + \frac{4}{- 1}$

Step 7.1.4.3.1.3

Multiply $2$ by $- 1$ .

$| x - 2 | = - 2 x + \frac{4}{- 1}$

Step 7.1.4.3.1.4

Divide $4$ by $- 1$ .

$| x - 2 | = - 2 x - 4$

Step 7.2

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

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

Step 7.3

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

$x - 2 = - 2 x - 4$

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

Step 7.4

Solve $x - 2 = - 2 x - 4$ 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

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

$x - 2 + 2 x = - 4$

Step 7.4.1.2

Add $x$ and $2 x$ .

$3 x - 2 = - 4$

Step 7.4.2

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

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Step 7.4.2.1

Add $2$ to both sides of the equation.

$3 x = - 4 + 2$

Step 7.4.2.2

Add $- 4$ and $2$ .

$3 x = - 2$

Step 7.4.3

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

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Step 7.4.3.1

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

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

Step 7.4.3.2

Simplify the left side.

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Step 7.4.3.2.1

Cancel the common factor of $3$ .

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Step 7.4.3.2.1.1

Cancel the common factor.

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

Step 7.4.3.2.1.2

Divide $x$ by $1$ .

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

Step 7.4.3.3

Simplify the right side.

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Step 7.4.3.3.1

Move the negative in front of the fraction.

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

Step 7.5

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

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Step 7.5.1

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

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Step 7.5.1.1

Rewrite.

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

Step 7.5.1.2

Simplify by adding zeros.

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

Step 7.5.1.3

Apply the distributive property.

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

Step 7.5.1.4

Multiply.

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Step 7.5.1.4.1

Multiply $- 2$ by $- 1$ .

$x - 2 = 2 x - - 4$

Step 7.5.1.4.2

Multiply $- 1$ by $- 4$ .

$x - 2 = 2 x + 4$

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

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

$x - 2 - 2 x = 4$

Step 7.5.2.2

Subtract $2 x$ from $x$ .

$- x - 2 = 4$

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

Add $2$ to both sides of the equation.

$- x = 4 + 2$

Step 7.5.3.2

Add $4$ and $2$ .

$- x = 6$

Step 7.5.4

Divide each term in $- x = 6$ by $- 1$ and simplify.

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Step 7.5.4.1

Divide each term in $- x = 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{6}{- 1}$

Step 7.5.4.2

Simplify the left side.

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Step 7.5.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{6}{- 1}$

Step 7.5.4.2.2

Divide $x$ by $1$ .

$x = \frac{6}{- 1}$

Step 7.5.4.3

Simplify the right side.

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Step 7.5.4.3.1

Divide $6$ by $- 1$ .

$x = - 6$

Step 7.6

Consolidate the solutions.

$x = - \frac{2}{3}, - 6$

Step 8

Consolidate the solutions.

$x = 12, - 8, - \frac{2}{3}, - 6$

Step 9

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

$x = 12$

Step 10