Algebra Examples

$| x + 1 | + | 2 x + 3 | = 5$

Step 1

Subtract $| x + 1 |$ from both sides of the equation.

$| 2 x + 3 | = 5 - | x + 1 |$

Step 2

Solve for $| x + 1 |$ .

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Step 2.1

Rewrite the equation as $5 - | x + 1 | = | 2 x + 3 |$ .

$5 - | x + 1 | = | 2 x + 3 |$

Step 2.2

Subtract $5$ from both sides of the equation.

$- | x + 1 | = | 2 x + 3 | - 5$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2

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

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Move the negative one from the denominator of $\frac{| 2 x + 3 |}{- 1}$ .

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

Step 2.3.3.1.2

Rewrite $- 1 \cdot | 2 x + 3 |$ as $- | 2 x + 3 |$ .

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

Step 2.3.3.1.3

Divide $- 5$ by $- 1$ .

$| x + 1 | = - | 2 x + 3 | + 5$

Step 3

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

$x + 1 = \pm (- | 2 x + 3 | + 5)$

Step 4

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

$x + 1 = - | 2 x + 3 | + 5$

$x + 1 = - (- | 2 x + 3 | + 5)$

Step 5

Solve $x + 1 = - | 2 x + 3 | + 5$ for $x$ .

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Step 5.1

Solve for $| 2 x + 3 |$ .

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Step 5.1.1

Rewrite the equation as $- | 2 x + 3 | + 5 = x + 1$ .

$- | 2 x + 3 | + 5 = x + 1$

Step 5.1.2

Move all terms not containing $| 2 x + 3 |$ to the right side of the equation.

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Step 5.1.2.1

Subtract $5$ from both sides of the equation.

$- | 2 x + 3 | = x + 1 - 5$

Step 5.1.2.2

Subtract $5$ from $1$ .

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

Step 5.1.3

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

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Step 5.1.3.1

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

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

Step 5.1.3.2

Simplify the left side.

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Step 5.1.3.2.1

Dividing two negative values results in a positive value.

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

Step 5.1.3.2.2

Divide $| 2 x + 3 |$ by $1$ .

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

Step 5.1.3.3

Simplify the right side.

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Step 5.1.3.3.1

Simplify each term.

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Step 5.1.3.3.1.1

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

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

Step 5.1.3.3.1.2

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

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

Step 5.1.3.3.1.3

Divide $- 4$ by $- 1$ .

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

Step 5.2

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

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

Step 5.3

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

$2 x + 3 = - x + 4$

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

Step 5.4

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

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Step 5.4.1

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

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Step 5.4.1.1

Add $x$ to both sides of the equation.

$2 x + 3 + x = 4$

Step 5.4.1.2

Add $2 x$ and $x$ .

$3 x + 3 = 4$

Step 5.4.2

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

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Step 5.4.2.1

Subtract $3$ from both sides of the equation.

$3 x = 4 - 3$

Step 5.4.2.2

Subtract $3$ from $4$ .

$3 x = 1$

Step 5.4.3

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 5.4.3.1

Divide each term in $3 x = 1$ by $3$ .

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

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $3$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

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

Step 5.4.3.2.1.2

Divide $x$ by $1$ .

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

Step 5.5

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

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Step 5.5.1

Simplify $- (- x + 4)$ .

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Step 5.5.1.1

Rewrite.

$2 x + 3 = 0 + 0 - (- x + 4)$

Step 5.5.1.2

Simplify by adding zeros.

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

Step 5.5.1.3

Apply the distributive property.

$2 x + 3 = - - x - 1 \cdot 4$

Step 5.5.1.4

Multiply $- - x$ .

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Step 5.5.1.4.1

Multiply $- 1$ by $- 1$ .

$2 x + 3 = 1 x - 1 \cdot 4$

Step 5.5.1.4.2

Multiply $x$ by $1$ .

$2 x + 3 = x - 1 \cdot 4$

Step 5.5.1.5

Multiply $- 1$ by $4$ .

$2 x + 3 = x - 4$

Step 5.5.2

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

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Step 5.5.2.1

Subtract $x$ from both sides of the equation.

$2 x + 3 - x = - 4$

Step 5.5.2.2

Subtract $x$ from $2 x$ .

$x + 3 = - 4$

Step 5.5.3

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

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Step 5.5.3.1

Subtract $3$ from both sides of the equation.

$x = - 4 - 3$

Step 5.5.3.2

Subtract $3$ from $- 4$ .

$x = - 7$

Step 5.6

Consolidate the solutions.

$x = \frac{1}{3}, - 7$

Step 6

Solve $x + 1 = - (- | 2 x + 3 | + 5)$ for $x$ .

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Step 6.1

Solve for $| 2 x + 3 |$ .

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Step 6.1.1

Rewrite the equation as $- (- | 2 x + 3 | + 5) = x + 1$ .

$- (- | 2 x + 3 | + 5) = x + 1$

Step 6.1.2

Simplify $- (- | 2 x + 3 | + 5)$ .

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Step 6.1.2.1

Apply the distributive property.

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

Step 6.1.2.2

Multiply $- - | 2 x + 3 |$ .

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Step 6.1.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.2.2.2

Multiply $| 2 x + 3 |$ by $1$ .

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

Step 6.1.2.3

Multiply $- 1$ by $5$ .

$| 2 x + 3 | - 5 = x + 1$

Step 6.1.3

Move all terms not containing $| 2 x + 3 |$ to the right side of the equation.

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Step 6.1.3.1

Add $5$ to both sides of the equation.

$| 2 x + 3 | = x + 1 + 5$

Step 6.1.3.2

Add $1$ and $5$ .

$| 2 x + 3 | = x + 6$

Step 6.2

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

$2 x + 3 = \pm (x + 6)$

Step 6.3

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

$2 x + 3 = x + 6$

$2 x + 3 = - (x + 6)$

Step 6.4

Solve $2 x + 3 = x + 6$ for $x$ .

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Step 6.4.1

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

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Step 6.4.1.1

Subtract $x$ from both sides of the equation.

$2 x + 3 - x = 6$

Step 6.4.1.2

Subtract $x$ from $2 x$ .

$x + 3 = 6$

Step 6.4.2

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

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Step 6.4.2.1

Subtract $3$ from both sides of the equation.

$x = 6 - 3$

Step 6.4.2.2

Subtract $3$ from $6$ .

$x = 3$

Step 6.5

Solve $2 x + 3 = - (x + 6)$ for $x$ .

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Step 6.5.1

Simplify $- (x + 6)$ .

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Step 6.5.1.1

Rewrite.

$2 x + 3 = 0 + 0 - (x + 6)$

Step 6.5.1.2

Simplify by adding zeros.

$2 x + 3 = - (x + 6)$

Step 6.5.1.3

Apply the distributive property.

$2 x + 3 = - x - 1 \cdot 6$

Step 6.5.1.4

Multiply $- 1$ by $6$ .

$2 x + 3 = - x - 6$

Step 6.5.2

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

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Step 6.5.2.1

Add $x$ to both sides of the equation.

$2 x + 3 + x = - 6$

Step 6.5.2.2

Add $2 x$ and $x$ .

$3 x + 3 = - 6$

Step 6.5.3

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

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Step 6.5.3.1

Subtract $3$ from both sides of the equation.

$3 x = - 6 - 3$

Step 6.5.3.2

Subtract $3$ from $- 6$ .

$3 x = - 9$

Step 6.5.4

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

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Step 6.5.4.1

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

$\frac{3 x}{3} = \frac{- 9}{3}$

Step 6.5.4.2

Simplify the left side.

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Step 6.5.4.2.1

Cancel the common factor of $3$ .

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Step 6.5.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 9}{3}$

Step 6.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{3}$

Step 6.5.4.3

Simplify the right side.

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Step 6.5.4.3.1

Divide $- 9$ by $3$ .

$x = - 3$

Step 6.6

Consolidate the solutions.

$x = 3, - 3$

Step 7

Consolidate the solutions.

$x = \frac{1}{3}, - 7, 3, - 3$

Step 8

Exclude the solutions that do not make $| x + 1 | + | 2 x + 3 | = 5$ true.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{3}, - 3$

Step 10