Algebra Examples

$| 2 x - 5 | = 4 - | x - 3 |$

Step 1

Rewrite the equation as $4 - | x - 3 | = | 2 x - 5 |$ .

$4 - | x - 3 | = | 2 x - 5 |$

Step 2

Subtract $4$ from both sides of the equation.

$- | x - 3 | = | 2 x - 5 | - 4$

Step 3

Divide each term in $- | x - 3 | = | 2 x - 5 | - 4$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- | x - 3 | = | 2 x - 5 | - 4$ by $- 1$ .

$\frac{- | x - 3 |}{- 1} = \frac{| 2 x - 5 |}{- 1} + \frac{- 4}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{| x - 3 |}{1} = \frac{| 2 x - 5 |}{- 1} + \frac{- 4}{- 1}$

Step 3.2.2

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

$| x - 3 | = \frac{| 2 x - 5 |}{- 1} + \frac{- 4}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative one from the denominator of $\frac{| 2 x - 5 |}{- 1}$ .

$| x - 3 | = - 1 \cdot | 2 x - 5 | + \frac{- 4}{- 1}$

Step 3.3.1.2

Rewrite $- 1 \cdot | 2 x - 5 |$ as $- | 2 x - 5 |$ .

$| x - 3 | = - | 2 x - 5 | + \frac{- 4}{- 1}$

Step 3.3.1.3

Divide $- 4$ by $- 1$ .

$| x - 3 | = - | 2 x - 5 | + 4$

Step 4

Solve for $| 2 x - 5 |$ .

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Step 4.1

Rewrite the equation as $- | 2 x - 5 | + 4 = | x - 3 |$ .

$- | 2 x - 5 | + 4 = | x - 3 |$

Step 4.2

Subtract $4$ from both sides of the equation.

$- | 2 x - 5 | = | x - 3 | - 4$

Step 4.3

Divide each term in $- | 2 x - 5 | = | x - 3 | - 4$ by $- 1$ and simplify.

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Step 4.3.1

Divide each term in $- | 2 x - 5 | = | x - 3 | - 4$ by $- 1$ .

$\frac{- | 2 x - 5 |}{- 1} = \frac{| x - 3 |}{- 1} + \frac{- 4}{- 1}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{| 2 x - 5 |}{1} = \frac{| x - 3 |}{- 1} + \frac{- 4}{- 1}$

Step 4.3.2.2

Divide $| 2 x - 5 |$ by $1$ .

$| 2 x - 5 | = \frac{| x - 3 |}{- 1} + \frac{- 4}{- 1}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

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

$| 2 x - 5 | = - 1 \cdot | x - 3 | + \frac{- 4}{- 1}$

Step 4.3.3.1.2

Rewrite $- 1 \cdot | x - 3 |$ as $- | x - 3 |$ .

$| 2 x - 5 | = - | x - 3 | + \frac{- 4}{- 1}$

Step 4.3.3.1.3

Divide $- 4$ by $- 1$ .

$| 2 x - 5 | = - | x - 3 | + 4$

Step 5

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

$2 x - 5 = \pm (- | x - 3 | + 4)$

Step 6

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

$2 x - 5 = - | x - 3 | + 4$

$2 x - 5 = - (- | x - 3 | + 4)$

Step 7

Solve $2 x - 5 = - | x - 3 | + 4$ for $x$ .

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Step 7.1

Solve for $| x - 3 |$ .

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Step 7.1.1

Rewrite the equation as $- | x - 3 | + 4 = 2 x - 5$ .

$- | x - 3 | + 4 = 2 x - 5$

Step 7.1.2

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

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Step 7.1.2.1

Subtract $4$ from both sides of the equation.

$- | x - 3 | = 2 x - 5 - 4$

Step 7.1.2.2

Subtract $4$ from $- 5$ .

$- | x - 3 | = 2 x - 9$

Step 7.1.3

Divide each term in $- | x - 3 | = 2 x - 9$ by $- 1$ and simplify.

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Step 7.1.3.1

Divide each term in $- | x - 3 | = 2 x - 9$ by $- 1$ .

$\frac{- | x - 3 |}{- 1} = \frac{2 x}{- 1} + \frac{- 9}{- 1}$

Step 7.1.3.2

Simplify the left side.

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Step 7.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{| x - 3 |}{1} = \frac{2 x}{- 1} + \frac{- 9}{- 1}$

Step 7.1.3.2.2

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

$| x - 3 | = \frac{2 x}{- 1} + \frac{- 9}{- 1}$

Step 7.1.3.3

Simplify the right side.

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Step 7.1.3.3.1

Simplify each term.

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Step 7.1.3.3.1.1

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

$| x - 3 | = - 1 \cdot (2 x) + \frac{- 9}{- 1}$

Step 7.1.3.3.1.2

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

$| x - 3 | = - (2 x) + \frac{- 9}{- 1}$

Step 7.1.3.3.1.3

Multiply $2$ by $- 1$ .

$| x - 3 | = - 2 x + \frac{- 9}{- 1}$

Step 7.1.3.3.1.4

Divide $- 9$ by $- 1$ .

$| x - 3 | = - 2 x + 9$

Step 7.2

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

$x - 3 = \pm (- 2 x + 9)$

Step 7.3

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

$x - 3 = - 2 x + 9$

$x - 3 = - (- 2 x + 9)$

Step 7.4

Solve $x - 3 = - 2 x + 9$ 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.

$x - 3 + 2 x = 9$

Step 7.4.1.2

Add $x$ and $2 x$ .

$3 x - 3 = 9$

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 $3$ to both sides of the equation.

$3 x = 9 + 3$

Step 7.4.2.2

Add $9$ and $3$ .

$3 x = 12$

Step 7.4.3

Divide each term in $3 x = 12$ by $3$ and simplify.

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Step 7.4.3.1

Divide each term in $3 x = 12$ by $3$ .

$\frac{3 x}{3} = \frac{12}{3}$

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.

$\frac{3 x}{3} = \frac{12}{3}$

Step 7.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{3}$

Step 7.4.3.3

Simplify the right side.

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Step 7.4.3.3.1

Divide $12$ by $3$ .

$x = 4$

Step 7.5

Solve $x - 3 = - (- 2 x + 9)$ for $x$ .

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Step 7.5.1

Simplify $- (- 2 x + 9)$ .

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Step 7.5.1.1

Rewrite.

$x - 3 = 0 + 0 - (- 2 x + 9)$

Step 7.5.1.2

Simplify by adding zeros.

$x - 3 = - (- 2 x + 9)$

Step 7.5.1.3

Apply the distributive property.

$x - 3 = - (- 2 x) - 1 \cdot 9$

Step 7.5.1.4

Multiply.

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Step 7.5.1.4.1

Multiply $- 2$ by $- 1$ .

$x - 3 = 2 x - 1 \cdot 9$

Step 7.5.1.4.2

Multiply $- 1$ by $9$ .

$x - 3 = 2 x - 9$

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.

$x - 3 - 2 x = - 9$

Step 7.5.2.2

Subtract $2 x$ from $x$ .

$- x - 3 = - 9$

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 $3$ to both sides of the equation.

$- x = - 9 + 3$

Step 7.5.3.2

Add $- 9$ and $3$ .

$- x = - 6$

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.

$x = 4, 6$

Step 8

Solve $2 x - 5 = - (- | x - 3 | + 4)$ for $x$ .

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Step 8.1

Solve for $| x - 3 |$ .

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Step 8.1.1

Rewrite the equation as $- (- | x - 3 | + 4) = 2 x - 5$ .

$- (- | x - 3 | + 4) = 2 x - 5$

Step 8.1.2

Simplify $- (- | x - 3 | + 4)$ .

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Step 8.1.2.1

Apply the distributive property.

$- - | x - 3 | - 1 \cdot 4 = 2 x - 5$

Step 8.1.2.2

Multiply $- - | x - 3 |$ .

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Step 8.1.2.2.1

Multiply $- 1$ by $- 1$ .

$1 | x - 3 | - 1 \cdot 4 = 2 x - 5$

Step 8.1.2.2.2

Multiply $| x - 3 |$ by $1$ .

$| x - 3 | - 1 \cdot 4 = 2 x - 5$

Step 8.1.2.3

Multiply $- 1$ by $4$ .

$| x - 3 | - 4 = 2 x - 5$

Step 8.1.3

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

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Step 8.1.3.1

Add $4$ to both sides of the equation.

$| x - 3 | = 2 x - 5 + 4$

Step 8.1.3.2

Add $- 5$ and $4$ .

$| x - 3 | = 2 x - 1$

Step 8.2

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

$x - 3 = \pm (2 x - 1)$

Step 8.3

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

$x - 3 = 2 x - 1$

$x - 3 = - (2 x - 1)$

Step 8.4

Solve $x - 3 = 2 x - 1$ for $x$ .

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Step 8.4.1

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

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Step 8.4.1.1

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

$x - 3 - 2 x = - 1$

Step 8.4.1.2

Subtract $2 x$ from $x$ .

$- x - 3 = - 1$

Step 8.4.2

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

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Step 8.4.2.1

Add $3$ to both sides of the equation.

$- x = - 1 + 3$

Step 8.4.2.2

Add $- 1$ and $3$ .

$- x = 2$

Step 8.4.3

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

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Step 8.4.3.1

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

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

Step 8.4.3.2

Simplify the left side.

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Step 8.4.3.2.1

Dividing two negative values results in a positive value.

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

Step 8.4.3.2.2

Divide $x$ by $1$ .

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

Step 8.4.3.3

Simplify the right side.

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Step 8.4.3.3.1

Divide $2$ by $- 1$ .

$x = - 2$

Step 8.5

Solve $x - 3 = - (2 x - 1)$ for $x$ .

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Step 8.5.1

Simplify $- (2 x - 1)$ .

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Step 8.5.1.1

Rewrite.

$x - 3 = 0 + 0 - (2 x - 1)$

Step 8.5.1.2

Simplify by adding zeros.

$x - 3 = - (2 x - 1)$

Step 8.5.1.3

Apply the distributive property.

$x - 3 = - (2 x) - - 1$

Step 8.5.1.4

Multiply.

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Step 8.5.1.4.1

Multiply $2$ by $- 1$ .

$x - 3 = - 2 x - - 1$

Step 8.5.1.4.2

Multiply $- 1$ by $- 1$ .

$x - 3 = - 2 x + 1$

Step 8.5.2

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

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Step 8.5.2.1

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

$x - 3 + 2 x = 1$

Step 8.5.2.2

Add $x$ and $2 x$ .

$3 x - 3 = 1$

Step 8.5.3

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

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Step 8.5.3.1

Add $3$ to both sides of the equation.

$3 x = 1 + 3$

Step 8.5.3.2

Add $1$ and $3$ .

$3 x = 4$

Step 8.5.4

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 8.5.4.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 8.5.4.2

Simplify the left side.

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Step 8.5.4.2.1

Cancel the common factor of $3$ .

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Step 8.5.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 8.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 8.6

Consolidate the solutions.

$x = - 2, \frac{4}{3}$

Step 9

Consolidate the solutions.

$x = 4, 6, - 2, \frac{4}{3}$

Step 10

Exclude the solutions that do not make $| 2 x - 5 | = 4 - | x - 3 |$ true.

$x = 4, \frac{4}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = 4, \frac{4}{3}$

Decimal Form:

$x = 4, 1.\overline{3}$

Mixed Number Form:

$x = 4, 1 \frac{1}{3}$

Step 12