Basic Math Examples

$| a - 3 | - | a - 2 | = 1$

Step 1

Subtract $| a - 3 |$ from both sides of the equation.

$- | a - 2 | = 1 - | a - 3 |$

Step 2

Divide each term in $- | a - 2 | = 1 - | a - 3 |$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- | a - 2 | = 1 - | a - 3 |$ by $- 1$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2

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

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

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

Divide $1$ by $- 1$ .

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

Step 2.3.1.2

Dividing two negative values results in a positive value.

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

Step 2.3.1.3

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

$| a - 2 | = - 1 + | a - 3 |$

Step 3

Solve for $| a - 3 |$ .

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Step 3.1

Rewrite the equation as $- 1 + | a - 3 | = | a - 2 |$ .

$- 1 + | a - 3 | = | a - 2 |$

Step 3.2

Add $1$ to both sides of the equation.

$| a - 3 | = | a - 2 | + 1$

Step 4

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

$a - 3 = \pm (| a - 2 | + 1)$

Step 5

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

$a - 3 = | a - 2 | + 1$

$a - 3 = - (| a - 2 | + 1)$

Step 6

Solve $a - 3 = | a - 2 | + 1$ for $a$ .

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Step 6.1

Solve for $| a - 2 |$ .

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Step 6.1.1

Rewrite the equation as $| a - 2 | + 1 = a - 3$ .

$| a - 2 | + 1 = a - 3$

Step 6.1.2

Move all terms not containing $| a - 2 |$ to the right side of the equation.

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Step 6.1.2.1

Subtract $1$ from both sides of the equation.

$| a - 2 | = a - 3 - 1$

Step 6.1.2.2

Subtract $1$ from $- 3$ .

$| a - 2 | = a - 4$

Step 6.2

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

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

Step 6.3

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

$a - 2 = a - 4$

$a - 2 = - (a - 4)$

Step 6.4

Solve $a - 2 = a - 4$ for $a$ .

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Step 6.4.1

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

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Step 6.4.1.1

Subtract $a$ from both sides of the equation.

$a - 2 - a = - 4$

Step 6.4.1.2

Combine the opposite terms in $a - 2 - a$ .

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Step 6.4.1.2.1

Subtract $a$ from $a$ .

$0 - 2 = - 4$

Step 6.4.1.2.2

Subtract $2$ from $0$ .

$- 2 = - 4$

Step 6.4.2

Since $- 2 \neq - 4$ , there are no solutions.

No solution

Step 6.5

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

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Step 6.5.1

Simplify $- (a - 4)$ .

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Step 6.5.1.1

Rewrite.

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

Step 6.5.1.2

Simplify by adding zeros.

$a - 2 = - (a - 4)$

Step 6.5.1.3

Apply the distributive property.

$a - 2 = - a - - 4$

Step 6.5.1.4

Multiply $- 1$ by $- 4$ .

$a - 2 = - a + 4$

Step 6.5.2

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

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Step 6.5.2.1

Add $a$ to both sides of the equation.

$a - 2 + a = 4$

Step 6.5.2.2

Add $a$ and $a$ .

$2 a - 2 = 4$

Step 6.5.3

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

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Step 6.5.3.1

Add $2$ to both sides of the equation.

$2 a = 4 + 2$

Step 6.5.3.2

Add $4$ and $2$ .

$2 a = 6$

Step 6.5.4

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

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Step 6.5.4.1

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

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

Step 6.5.4.2

Simplify the left side.

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Step 6.5.4.2.1

Cancel the common factor of $2$ .

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Step 6.5.4.2.1.1

Cancel the common factor.

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

Step 6.5.4.2.1.2

Divide $a$ by $1$ .

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

Step 6.5.4.3

Simplify the right side.

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Step 6.5.4.3.1

Divide $6$ by $2$ .

$a = 3$

Step 6.6

Consolidate the solutions.

$a = 3$

Step 7

Solve $a - 3 = - (| a - 2 | + 1)$ for $a$ .

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Step 7.1

Solve for $| a - 2 |$ .

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Step 7.1.1

Rewrite the equation as $- (| a - 2 | + 1) = a - 3$ .

$- (| a - 2 | + 1) = a - 3$

Step 7.1.2

Simplify $- (| a - 2 | + 1)$ .

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Step 7.1.2.1

Apply the distributive property.

$- | a - 2 | - 1 \cdot 1 = a - 3$

Step 7.1.2.2

Multiply $- 1$ by $1$ .

$- | a - 2 | - 1 = a - 3$

Step 7.1.3

Move all terms not containing $| a - 2 |$ to the right side of the equation.

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Step 7.1.3.1

Add $1$ to both sides of the equation.

$- | a - 2 | = a - 3 + 1$

Step 7.1.3.2

Add $- 3$ and $1$ .

$- | a - 2 | = a - 2$

Step 7.1.4

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

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Step 7.1.4.1

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

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

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.

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

Step 7.1.4.2.2

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

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

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

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

Step 7.1.4.3.1.2

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

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

Step 7.1.4.3.1.3

Divide $- 2$ by $- 1$ .

$| a - 2 | = - a + 2$

Step 7.2

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

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

Step 7.3

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

$a - 2 = - a + 2$

$a - 2 = - (- a + 2)$

Step 7.4

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

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Step 7.4.1

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

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Step 7.4.1.1

Add $a$ to both sides of the equation.

$a - 2 + a = 2$

Step 7.4.1.2

Add $a$ and $a$ .

$2 a - 2 = 2$

Step 7.4.2

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

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Step 7.4.2.1

Add $2$ to both sides of the equation.

$2 a = 2 + 2$

Step 7.4.2.2

Add $2$ and $2$ .

$2 a = 4$

Step 7.4.3

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

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Step 7.4.3.1

Divide each term in $2 a = 4$ by $2$ .

$\frac{2 a}{2} = \frac{4}{2}$

Step 7.4.3.2

Simplify the left side.

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Step 7.4.3.2.1

Cancel the common factor of $2$ .

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Step 7.4.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{4}{2}$

Step 7.4.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{4}{2}$

Step 7.4.3.3

Simplify the right side.

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Step 7.4.3.3.1

Divide $4$ by $2$ .

$a = 2$

Step 7.5

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

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Step 7.5.1

Simplify $- (- a + 2)$ .

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Step 7.5.1.1

Rewrite.

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

Step 7.5.1.2

Simplify by adding zeros.

$a - 2 = - (- a + 2)$

Step 7.5.1.3

Apply the distributive property.

$a - 2 = - - a - 1 \cdot 2$

Step 7.5.1.4

Multiply $- - a$ .

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Step 7.5.1.4.1

Multiply $- 1$ by $- 1$ .

$a - 2 = 1 a - 1 \cdot 2$

Step 7.5.1.4.2

Multiply $a$ by $1$ .

$a - 2 = a - 1 \cdot 2$

Step 7.5.1.5

Multiply $- 1$ by $2$ .

$a - 2 = a - 2$

Step 7.5.2

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

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Step 7.5.2.1

Subtract $a$ from both sides of the equation.

$a - 2 - a = - 2$

Step 7.5.2.2

Combine the opposite terms in $a - 2 - a$ .

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Step 7.5.2.2.1

Subtract $a$ from $a$ .

$0 - 2 = - 2$

Step 7.5.2.2.2

Subtract $2$ from $0$ .

$- 2 = - 2$

Step 7.5.3

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

All real numbers

Step 7.6

Consolidate the solutions.

$a = 2$

Step 8

Consolidate the solutions.

$a = 3, 2$

Step 9

The solution consists of all of the true intervals.

$a \leq 2$

Step 10

Exclude the solutions that do not make $| a - 3 | - | a - 2 | = 1$ true.

$a = 2$

Step 11