Algebra Examples

$| 2 x - 5 | = 3 + 2 | 7 - x |$

Step 1

Rewrite the equation as $3 + 2 | 7 - x | = | 2 x - 5 |$ .

$3 + 2 | 7 - x | = | 2 x - 5 |$

Step 2

Subtract $3$ from both sides of the equation.

$2 | 7 - x | = | 2 x - 5 | - 3$

Step 3

Divide each term in $2 | 7 - x | = | 2 x - 5 | - 3$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 | 7 - x | = | 2 x - 5 | - 3$ by $2$ .

$\frac{2 | 7 - x |}{2} = \frac{| 2 x - 5 |}{2} + \frac{- 3}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 | 7 - x |}{2} = \frac{| 2 x - 5 |}{2} + \frac{- 3}{2}$

Step 3.2.1.2

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

$| 7 - x | = \frac{| 2 x - 5 |}{2} + \frac{- 3}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$| 7 - x | = \frac{| 2 x - 5 |}{2} - \frac{3}{2}$

Step 4

Solve for $| 2 x - 5 |$ .

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Step 4.1

Rewrite the equation as $\frac{| 2 x - 5 |}{2} - \frac{3}{2} = | 7 - x |$ .

$\frac{| 2 x - 5 |}{2} - \frac{3}{2} = | 7 - x |$

Step 4.2

Add $\frac{3}{2}$ to both sides of the equation.

$\frac{| 2 x - 5 |}{2} = | 7 - x | + \frac{3}{2}$

Step 4.3

Multiply both sides of the equation by $2$ .

$2 \frac{| 2 x - 5 |}{2} = 2 (| 7 - x | + \frac{3}{2})$

Step 4.4

Simplify both sides of the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Cancel the common factor of $2$ .

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Step 4.4.1.1.1

Cancel the common factor.

$2 \frac{| 2 x - 5 |}{2} = 2 (| 7 - x | + \frac{3}{2})$

Step 4.4.1.1.2

Rewrite the expression.

$| 2 x - 5 | = 2 (| 7 - x | + \frac{3}{2})$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $2 (| 7 - x | + \frac{3}{2})$ .

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Step 4.4.2.1.1

Apply the distributive property.

$| 2 x - 5 | = 2 | 7 - x | + 2 (\frac{3}{2})$

Step 4.4.2.1.2

Cancel the common factor of $2$ .

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Step 4.4.2.1.2.1

Cancel the common factor.

$| 2 x - 5 | = 2 | 7 - x | + 2 (\frac{3}{2})$

Step 4.4.2.1.2.2

Rewrite the expression.

$| 2 x - 5 | = 2 | 7 - x | + 3$

Step 5

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

$2 x - 5 = \pm (2 | 7 - x | + 3)$

Step 6

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

$2 x - 5 = 2 | 7 - x | + 3$

$2 x - 5 = - (2 | 7 - x | + 3)$

Step 7

Solve $2 x - 5 = 2 | 7 - x | + 3$ for $x$ .

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Step 7.1

Solve for $| 7 - x |$ .

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Step 7.1.1

Rewrite the equation as $2 | 7 - x | + 3 = 2 x - 5$ .

$2 | 7 - x | + 3 = 2 x - 5$

Step 7.1.2

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

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Step 7.1.2.1

Subtract $3$ from both sides of the equation.

$2 | 7 - x | = 2 x - 5 - 3$

Step 7.1.2.2

Subtract $3$ from $- 5$ .

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

Step 7.1.3

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

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Step 7.1.3.1

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

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

Step 7.1.3.2

Simplify the left side.

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Step 7.1.3.2.1

Cancel the common factor of $2$ .

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Step 7.1.3.2.1.1

Cancel the common factor.

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

Step 7.1.3.2.1.2

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

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

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

Cancel the common factor of $2$ .

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Step 7.1.3.3.1.1.1

Cancel the common factor.

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

Step 7.1.3.3.1.1.2

Divide $x$ by $1$ .

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

Step 7.1.3.3.1.2

Divide $- 8$ by $2$ .

$| 7 - x | = x - 4$

Step 7.2

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

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

Step 7.3

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

$7 - x = x - 4$

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

Step 7.4

Solve $7 - x = x - 4$ 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

Subtract $x$ from both sides of the equation.

$7 - x - x = - 4$

Step 7.4.1.2

Subtract $x$ from $- x$ .

$7 - 2 x = - 4$

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

Subtract $7$ from both sides of the equation.

$- 2 x = - 4 - 7$

Step 7.4.2.2

Subtract $7$ from $- 4$ .

$- 2 x = - 11$

Step 7.4.3

Divide each term in $- 2 x = - 11$ by $- 2$ and simplify.

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Step 7.4.3.1

Divide each term in $- 2 x = - 11$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 11}{- 2}$

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.

$\frac{- 2 x}{- 2} = \frac{- 11}{- 2}$

Step 7.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{- 2}$

Step 7.4.3.3

Simplify the right side.

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Step 7.4.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{11}{2}$

Step 7.5

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

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Step 7.5.1

Simplify $- (x - 4)$ .

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Step 7.5.1.1

Rewrite.

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

Step 7.5.1.2

Simplify by adding zeros.

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

Step 7.5.1.3

Apply the distributive property.

$7 - x = - x - - 4$

Step 7.5.1.4

Multiply $- 1$ by $- 4$ .

$7 - x = - x + 4$

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

Add $x$ to both sides of the equation.

$7 - x + x = 4$

Step 7.5.2.2

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

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Step 7.5.2.2.1

Add $- x$ and $x$ .

$7 + 0 = 4$

Step 7.5.2.2.2

Add $7$ and $0$ .

$7 = 4$

Step 7.5.3

Since $7 \neq 4$ , there are no solutions.

No solution

Step 7.6

Consolidate the solutions.

$x = \frac{11}{2}$

Step 8

Solve $2 x - 5 = - (2 | 7 - x | + 3)$ for $x$ .

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Step 8.1

Solve for $| 7 - x |$ .

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Step 8.1.1

Rewrite the equation as $- (2 | 7 - x | + 3) = 2 x - 5$ .

$- (2 | 7 - x | + 3) = 2 x - 5$

Step 8.1.2

Simplify $- (2 | 7 - x | + 3)$ .

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Step 8.1.2.1

Apply the distributive property.

$- (2 | 7 - x |) - 1 \cdot 3 = 2 x - 5$

Step 8.1.2.2

Multiply.

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Step 8.1.2.2.1

Multiply $2$ by $- 1$ .

$- 2 | 7 - x | - 1 \cdot 3 = 2 x - 5$

Step 8.1.2.2.2

Multiply $- 1$ by $3$ .

$- 2 | 7 - x | - 3 = 2 x - 5$

Step 8.1.3

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

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Step 8.1.3.1

Add $3$ to both sides of the equation.

$- 2 | 7 - x | = 2 x - 5 + 3$

Step 8.1.3.2

Add $- 5$ and $3$ .

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

Step 8.1.4

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

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Step 8.1.4.1

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

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

Step 8.1.4.2

Simplify the left side.

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Step 8.1.4.2.1

Cancel the common factor of $- 2$ .

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Step 8.1.4.2.1.1

Cancel the common factor.

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

Step 8.1.4.2.1.2

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

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

Step 8.1.4.3

Simplify the right side.

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Step 8.1.4.3.1

Simplify each term.

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Step 8.1.4.3.1.1

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

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Step 8.1.4.3.1.1.1

Factor $2$ out of $2 x$ .

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

Step 8.1.4.3.1.1.2

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

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

Step 8.1.4.3.1.2

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

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

Step 8.1.4.3.1.3

Divide $- 2$ by $- 2$ .

$| 7 - x | = - x + 1$

Step 8.2

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

$7 - x = \pm (- x + 1)$

Step 8.3

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

$7 - x = - x + 1$

$7 - x = - (- x + 1)$

Step 8.4

Solve $7 - x = - x + 1$ 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

Add $x$ to both sides of the equation.

$7 - x + x = 1$

Step 8.4.1.2

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

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Step 8.4.1.2.1

Add $- x$ and $x$ .

$7 + 0 = 1$

Step 8.4.1.2.2

Add $7$ and $0$ .

$7 = 1$

Step 8.4.2

Since $7 \neq 1$ , there are no solutions.

No solution

Step 8.5

Solve $7 - x = - (- x + 1)$ for $x$ .

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Step 8.5.1

Simplify $- (- x + 1)$ .

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Step 8.5.1.1

Rewrite.

$7 - x = 0 + 0 - (- x + 1)$

Step 8.5.1.2

Simplify by adding zeros.

$7 - x = - (- x + 1)$

Step 8.5.1.3

Apply the distributive property.

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

Step 8.5.1.4

Multiply $- - x$ .

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Step 8.5.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 8.5.1.4.2

Multiply $x$ by $1$ .

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

Step 8.5.1.5

Multiply $- 1$ by $1$ .

$7 - x = x - 1$

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

Subtract $x$ from both sides of the equation.

$7 - x - x = - 1$

Step 8.5.2.2

Subtract $x$ from $- x$ .

$7 - 2 x = - 1$

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

Subtract $7$ from both sides of the equation.

$- 2 x = - 1 - 7$

Step 8.5.3.2

Subtract $7$ from $- 1$ .

$- 2 x = - 8$

Step 8.5.4

Divide each term in $- 2 x = - 8$ by $- 2$ and simplify.

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Step 8.5.4.1

Divide each term in $- 2 x = - 8$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 8}{- 2}$

Step 8.5.4.2

Simplify the left side.

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Step 8.5.4.2.1

Cancel the common factor of $- 2$ .

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Step 8.5.4.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 8}{- 2}$

Step 8.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 2}$

Step 8.5.4.3

Simplify the right side.

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Step 8.5.4.3.1

Divide $- 8$ by $- 2$ .

$x = 4$

Step 8.6

Consolidate the solutions.

$x = 4$

Step 9

Consolidate the solutions.

$x = \frac{11}{2}, 4$

Step 10

Exclude the solutions that do not make $| 2 x - 5 | = 3 + 2 | 7 - x |$ true.

$x = \frac{11}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{2}$

Decimal Form:

$x = 5.5$

Mixed Number Form:

$x = 5 \frac{1}{2}$

Step 12