Basic Math Examples

$| a + b | = | a | + | b |$

Step 1

Rewrite the equation as $| a | + | b | = | a + b |$ .

$| a | + | b | = | a + b |$

Step 2

Subtract $| b |$ from both sides of the equation.

$| a | = | a + b | - | b |$

Step 3

Solve for $| b |$ .

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Step 3.1

Rewrite the equation as $| a + b | - | b | = | a |$ .

$| a + b | - | b | = | a |$

Step 3.2

Subtract $| a + b |$ from both sides of the equation.

$- | b | = | a | - | a + b |$

Step 3.3

Divide each term in $- | b | = | a | - | a + b |$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- | b | = | a | - | a + b |$ by $- 1$ .

$\frac{- | b |}{- 1} = \frac{| a |}{- 1} + \frac{- | a + b |}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{| b |}{1} = \frac{| a |}{- 1} + \frac{- | a + b |}{- 1}$

Step 3.3.2.2

Divide $| b |$ by $1$ .

$| b | = \frac{| a |}{- 1} + \frac{- | a + b |}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

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

$| b | = - 1 \cdot | a | + \frac{- | a + b |}{- 1}$

Step 3.3.3.1.2

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

$| b | = - | a | + \frac{- | a + b |}{- 1}$

Step 3.3.3.1.3

Dividing two negative values results in a positive value.

$| b | = - | a | + \frac{| a + b |}{1}$

Step 3.3.3.1.4

Divide $| a + b |$ by $1$ .

$| b | = - | a | + | a + b |$

Step 4

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

$b = \pm (- | a | + | a + b |)$

Step 5

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

$b = - | a | + | a + b |$

$b = - (- | a | + | a + b |)$

Step 6

Solve $b = - | a | + | a + b |$ for $a$ .

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Step 6.1

Solve for $| a + b |$ .

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Step 6.1.1

Rewrite the equation as $- | a | + | a + b | = b$ .

$- | a | + | a + b | = b$

Step 6.1.2

Add $| a |$ to both sides of the equation.

$| a + b | = b + | a |$

Step 6.2

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

$a + b = \pm (b + | a |)$

Step 6.3

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

$a + b = b + | a |$

$a + b = - (b + | a |)$

Step 6.4

Solve $a + b = b + | a |$ for $a$ .

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Step 6.4.1

Solve for $| a |$ .

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Step 6.4.1.1

Rewrite the equation as $b + | a | = a + b$ .

$b + | a | = a + b$

Step 6.4.1.2

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

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Step 6.4.1.2.1

Subtract $b$ from both sides of the equation.

$| a | = a + b - b$

Step 6.4.1.2.2

Combine the opposite terms in $a + b - b$ .

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Step 6.4.1.2.2.1

Subtract $b$ from $b$ .

$| a | = a + 0$

Step 6.4.1.2.2.2

Add $a$ and $0$ .

$| a | = a$

Step 6.4.2

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

$a = \pm (a)$

Step 6.4.3

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

$a = a$

$a = - (a)$

Step 6.4.4

Solve $a = a$ for $a$ .

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Step 6.4.4.1

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

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Step 6.4.4.1.1

Subtract $a$ from both sides of the equation.

$a - a = 0$

Step 6.4.4.1.2

Subtract $a$ from $a$ .

$0 = 0$

Step 6.4.4.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 6.4.5

Solve $a = - (a)$ for $a$ .

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Step 6.4.5.1

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

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Step 6.4.5.1.1

Add $a$ to both sides of the equation.

$a + a = 0$

Step 6.4.5.1.2

Add $a$ and $a$ .

$2 a = 0$

Step 6.4.5.2

Divide each term in $2 a = 0$ by $2$ and simplify.

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Step 6.4.5.2.1

Divide each term in $2 a = 0$ by $2$ .

$\frac{2 a}{2} = \frac{0}{2}$

Step 6.4.5.2.2

Simplify the left side.

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Step 6.4.5.2.2.1

Cancel the common factor of $2$ .

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Step 6.4.5.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{0}{2}$

Step 6.4.5.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0}{2}$

Step 6.4.5.2.3

Simplify the right side.

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Step 6.4.5.2.3.1

Divide $0$ by $2$ .

$a = 0$

Step 6.4.6

Consolidate the solutions.

$a = 0$

Step 6.5

Solve $a + b = - (b + | a |)$ for $a$ .

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Step 6.5.1

Solve for $| a |$ .

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Step 6.5.1.1

Rewrite the equation as $- (b + | a |) = a + b$ .

$- (b + | a |) = a + b$

Step 6.5.1.2

Apply the distributive property.

$- b - | a | = a + b$

Step 6.5.1.3

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

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Step 6.5.1.3.1

Add $b$ to both sides of the equation.

$- | a | = a + b + b$

Step 6.5.1.3.2

Add $b$ and $b$ .

$- | a | = a + 2 b$

Step 6.5.1.4

Divide each term in $- | a | = a + 2 b$ by $- 1$ and simplify.

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Step 6.5.1.4.1

Divide each term in $- | a | = a + 2 b$ by $- 1$ .

$\frac{- | a |}{- 1} = \frac{a}{- 1} + \frac{2 b}{- 1}$

Step 6.5.1.4.2

Simplify the left side.

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Step 6.5.1.4.2.1

Dividing two negative values results in a positive value.

$\frac{| a |}{1} = \frac{a}{- 1} + \frac{2 b}{- 1}$

Step 6.5.1.4.2.2

Divide $| a |$ by $1$ .

$| a | = \frac{a}{- 1} + \frac{2 b}{- 1}$

Step 6.5.1.4.3

Simplify the right side.

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Step 6.5.1.4.3.1

Simplify each term.

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Step 6.5.1.4.3.1.1

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

$| a | = - 1 \cdot a + \frac{2 b}{- 1}$

Step 6.5.1.4.3.1.2

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

$| a | = - a + \frac{2 b}{- 1}$

Step 6.5.1.4.3.1.3

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

$| a | = - a - 1 \cdot (2 b)$

Step 6.5.1.4.3.1.4

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

$| a | = - a - (2 b)$

Step 6.5.1.4.3.1.5

Multiply $2$ by $- 1$ .

$| a | = - a - 2 b$

Step 6.5.2

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

$a = \pm (- a - 2 b)$

Step 6.5.3

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

$a = - a - 2 b$

$a = - (- a - 2 b)$

Step 6.5.4

Solve $a = - a - 2 b$ for $a$ .

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Step 6.5.4.1

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

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Step 6.5.4.1.1

Add $a$ to both sides of the equation.

$a + a = - 2 b$

Step 6.5.4.1.2

Add $a$ and $a$ .

$2 a = - 2 b$

Step 6.5.4.2

Divide each term in $2 a = - 2 b$ by $2$ and simplify.

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Step 6.5.4.2.1

Divide each term in $2 a = - 2 b$ by $2$ .

$\frac{2 a}{2} = \frac{- 2 b}{2}$

Step 6.5.4.2.2

Simplify the left side.

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Step 6.5.4.2.2.1

Cancel the common factor of $2$ .

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Step 6.5.4.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 2 b}{2}$

Step 6.5.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 2 b}{2}$

Step 6.5.4.2.3

Simplify the right side.

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Step 6.5.4.2.3.1

Cancel the common factor of $- 2$ and $2$ .

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Step 6.5.4.2.3.1.1

Factor $2$ out of $- 2 b$ .

$a = \frac{2 (- b)}{2}$

Step 6.5.4.2.3.1.2

Cancel the common factors.

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Step 6.5.4.2.3.1.2.1

Factor $2$ out of $2$ .

$a = \frac{2 (- b)}{2 (1)}$

Step 6.5.4.2.3.1.2.2

Cancel the common factor.

$a = \frac{2 (- b)}{2 \cdot 1}$

Step 6.5.4.2.3.1.2.3

Rewrite the expression.

$a = \frac{- b}{1}$

Step 6.5.4.2.3.1.2.4

Divide $- b$ by $1$ .

$a = - b$

Step 6.5.5

Solve $a = - (- a - 2 b)$ for $a$ .

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Step 6.5.5.1

Simplify $- (- a - 2 b)$ .

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Step 6.5.5.1.1

Apply the distributive property.

$a = - - a - (- 2 b)$

Step 6.5.5.1.2

Multiply $- - a$ .

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Step 6.5.5.1.2.1

Multiply $- 1$ by $- 1$ .

$a = 1 a - (- 2 b)$

Step 6.5.5.1.2.2

Multiply $a$ by $1$ .

$a = a - (- 2 b)$

Step 6.5.5.1.3

Multiply $- 2$ by $- 1$ .

$a = a + 2 b$

Step 6.5.5.2

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

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Step 6.5.5.2.1

Subtract $a$ from both sides of the equation.

$a - a = 2 b$

Step 6.5.5.2.2

Subtract $a$ from $a$ .

$0 = 2 b$

Step 6.5.6

Consolidate the solutions.

$a = - b$

$a = 2 b$

$a = - b$

$a = 2 b$

Step 6.6

Consolidate the solutions.

$a = 0$

$a = - b$

$a = 2 b$

$a = 0$

$a = - b$

$a = 2 b$

Step 7

Solve $b = - (- | a | + | a + b |)$ for $a$ .

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Step 7.1

Solve for $| a + b |$ .

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Step 7.1.1

Rewrite the equation as $- (- | a | + | a + b |) = b$ .

$- (- | a | + | a + b |) = b$

Step 7.1.2

Simplify $- (- | a | + | a + b |)$ .

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Step 7.1.2.1

Apply the distributive property.

$- - | a | - | a + b | = b$

Step 7.1.2.2

Multiply $- - | a |$ .

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Step 7.1.2.2.1

Multiply $- 1$ by $- 1$ .

$1 | a | - | a + b | = b$

Step 7.1.2.2.2

Multiply $| a |$ by $1$ .

$| a | - | a + b | = b$

Step 7.1.3

Subtract $| a |$ from both sides of the equation.

$- | a + b | = b - | a |$

Step 7.1.4

Divide each term in $- | a + b | = b - | a |$ by $- 1$ and simplify.

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Step 7.1.4.1

Divide each term in $- | a + b | = b - | a |$ by $- 1$ .

$\frac{- | a + b |}{- 1} = \frac{b}{- 1} + \frac{- | a |}{- 1}$

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.

$\frac{| a + b |}{1} = \frac{b}{- 1} + \frac{- | a |}{- 1}$

Step 7.1.4.2.2

Divide $| a + b |$ by $1$ .

$| a + b | = \frac{b}{- 1} + \frac{- | a |}{- 1}$

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 $\frac{b}{- 1}$ .

$| a + b | = - 1 \cdot b + \frac{- | a |}{- 1}$

Step 7.1.4.3.1.2

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

$| a + b | = - b + \frac{- | a |}{- 1}$

Step 7.1.4.3.1.3

Dividing two negative values results in a positive value.

$| a + b | = - b + \frac{| a |}{1}$

Step 7.1.4.3.1.4

Divide $| a |$ by $1$ .

$| a + b | = - b + | a |$

Step 7.2

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

$a + b = \pm (- b + | a |)$

Step 7.3

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

$a + b = - b + | a |$

$a + b = - (- b + | a |)$

Step 7.4

Solve $a + b = - b + | a |$ for $a$ .

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Step 7.4.1

Solve for $| a |$ .

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Step 7.4.1.1

Rewrite the equation as $- b + | a | = a + b$ .

$- b + | a | = a + b$

Step 7.4.1.2

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

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Step 7.4.1.2.1

Add $b$ to both sides of the equation.

$| a | = a + b + b$

Step 7.4.1.2.2

Add $b$ and $b$ .

$| a | = a + 2 b$

Step 7.4.2

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

$a = \pm (a + 2 b)$

Step 7.4.3

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

$a = a + 2 b$

$a = - (a + 2 b)$

Step 7.4.4

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

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Step 7.4.4.1

Subtract $a$ from both sides of the equation.

$a - a = 2 b$

Step 7.4.4.2

Subtract $a$ from $a$ .

$0 = 2 b$

Step 7.4.5

Solve $a = - (a + 2 b)$ for $a$ .

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Step 7.4.5.1

Simplify $- (a + 2 b)$ .

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Step 7.4.5.1.1

Apply the distributive property.

$a = - a - (2 b)$

Step 7.4.5.1.2

Multiply $2$ by $- 1$ .

$a = - a - 2 b$

Step 7.4.5.2

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

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Step 7.4.5.2.1

Add $a$ to both sides of the equation.

$a + a = - 2 b$

Step 7.4.5.2.2

Add $a$ and $a$ .

$2 a = - 2 b$

Step 7.4.5.3

Divide each term in $2 a = - 2 b$ by $2$ and simplify.

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Step 7.4.5.3.1

Divide each term in $2 a = - 2 b$ by $2$ .

$\frac{2 a}{2} = \frac{- 2 b}{2}$

Step 7.4.5.3.2

Simplify the left side.

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Step 7.4.5.3.2.1

Cancel the common factor of $2$ .

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Step 7.4.5.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 2 b}{2}$

Step 7.4.5.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 2 b}{2}$

Step 7.4.5.3.3

Simplify the right side.

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Step 7.4.5.3.3.1

Cancel the common factor of $- 2$ and $2$ .

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Step 7.4.5.3.3.1.1

Factor $2$ out of $- 2 b$ .

$a = \frac{2 (- b)}{2}$

Step 7.4.5.3.3.1.2

Cancel the common factors.

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Step 7.4.5.3.3.1.2.1

Factor $2$ out of $2$ .

$a = \frac{2 (- b)}{2 (1)}$

Step 7.4.5.3.3.1.2.2

Cancel the common factor.

$a = \frac{2 (- b)}{2 \cdot 1}$

Step 7.4.5.3.3.1.2.3

Rewrite the expression.

$a = \frac{- b}{1}$

Step 7.4.5.3.3.1.2.4

Divide $- b$ by $1$ .

$a = - b$

Step 7.4.6

Consolidate the solutions.

$a = 2 b$

$a = - b$

$a = 2 b$

$a = - b$

Step 7.5

Solve $a + b = - (- b + | a |)$ for $a$ .

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Step 7.5.1

Solve for $| a |$ .

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Step 7.5.1.1

Rewrite the equation as $- (- b + | a |) = a + b$ .

$- (- b + | a |) = a + b$

Step 7.5.1.2

Simplify $- (- b + | a |)$ .

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Step 7.5.1.2.1

Apply the distributive property.

$- - b - | a | = a + b$

Step 7.5.1.2.2

Multiply $- - b$ .

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Step 7.5.1.2.2.1

Multiply $- 1$ by $- 1$ .

$1 b - | a | = a + b$

Step 7.5.1.2.2.2

Multiply $b$ by $1$ .

$b - | a | = a + b$

Step 7.5.1.3

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

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Step 7.5.1.3.1

Subtract $b$ from both sides of the equation.

$- | a | = a + b - b$

Step 7.5.1.3.2

Combine the opposite terms in $a + b - b$ .

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Step 7.5.1.3.2.1

Subtract $b$ from $b$ .

$- | a | = a + 0$

Step 7.5.1.3.2.2

Add $a$ and $0$ .

$- | a | = a$

Step 7.5.1.4

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

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Step 7.5.1.4.1

Divide each term in $- | a | = a$ by $- 1$ .

$\frac{- | a |}{- 1} = \frac{a}{- 1}$

Step 7.5.1.4.2

Simplify the left side.

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Step 7.5.1.4.2.1

Dividing two negative values results in a positive value.

$\frac{| a |}{1} = \frac{a}{- 1}$

Step 7.5.1.4.2.2

Divide $| a |$ by $1$ .

$| a | = \frac{a}{- 1}$

Step 7.5.1.4.3

Simplify the right side.

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Step 7.5.1.4.3.1

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

$| a | = - 1 \cdot a$

Step 7.5.1.4.3.2

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

$| a | = - a$

Step 7.5.2

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

$a = \pm (- a)$

Step 7.5.3

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

$a = - a$

$a = - (- a)$

Step 7.5.4

Solve $a = - a$ for $a$ .

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Step 7.5.4.1

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

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Step 7.5.4.1.1

Add $a$ to both sides of the equation.

$a + a = 0$

Step 7.5.4.1.2

Add $a$ and $a$ .

$2 a = 0$

Step 7.5.4.2

Divide each term in $2 a = 0$ by $2$ and simplify.

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Step 7.5.4.2.1

Divide each term in $2 a = 0$ by $2$ .

$\frac{2 a}{2} = \frac{0}{2}$

Step 7.5.4.2.2

Simplify the left side.

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Step 7.5.4.2.2.1

Cancel the common factor of $2$ .

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Step 7.5.4.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{0}{2}$

Step 7.5.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0}{2}$

Step 7.5.4.2.3

Simplify the right side.

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Step 7.5.4.2.3.1

Divide $0$ by $2$ .

$a = 0$

Step 7.5.5

Solve $a = - (- a)$ for $a$ .

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Step 7.5.5.1

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

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Step 7.5.5.1.1

Subtract $a$ from both sides of the equation.

$a - a = 0$

Step 7.5.5.1.2

Subtract $a$ from $a$ .

$0 = 0$

Step 7.5.5.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 7.5.6

Consolidate the solutions.

$a = 0$

Step 7.6

Consolidate the solutions.

$a = 2 b$

$a = - b$

$a = 0$

$a = 2 b$

$a = - b$

$a = 0$

Step 8

Consolidate the solutions.

$a = 0$

$a = - b$

$a = 2 b$