Finite Math Examples

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

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

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

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

Step 3

Solve for $| 3 x |$ .

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Step 3.1

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

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

Step 3.2

Subtract $| 4 x |$ from both sides of the equation.

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

Step 3.3

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

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Step 3.3.1

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

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

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.

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

Step 3.3.2.2

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

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

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 $\frac{| x + 3 |}{- 1}$ .

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

Step 3.3.3.1.2

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

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

Step 3.3.3.1.3

Dividing two negative values results in a positive value.

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

Step 3.3.3.1.4

Divide $| 4 x |$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 6

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

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Step 6.1

Solve for $| 4 x |$ .

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Step 6.1.1

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

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

Step 6.1.2

Add $| x + 3 |$ to both sides of the equation.

$| 4 x | = 3 x + | x + 3 |$

Step 6.2

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

$4 x = \pm (3 x + | x + 3 |)$

Step 6.3

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

$4 x = 3 x + | x + 3 |$

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

Step 6.4

Solve $4 x = 3 x + | x + 3 |$ for $x$ .

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Step 6.4.1

Solve for $| x + 3 |$ .

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Step 6.4.1.1

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

$3 x + | x + 3 | = 4 x$

Step 6.4.1.2

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

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Step 6.4.1.2.1

Subtract $3 x$ from both sides of the equation.

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

Step 6.4.1.2.2

Subtract $3 x$ from $4 x$ .

$| x + 3 | = x$

Step 6.4.2

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

$x + 3 = \pm (x)$

Step 6.4.3

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

$x + 3 = x$

$x + 3 = - (x)$

Step 6.4.4

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

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Step 6.4.4.1

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

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Step 6.4.4.1.1

Subtract $x$ from both sides of the equation.

$x + 3 - x = 0$

Step 6.4.4.1.2

Combine the opposite terms in $x + 3 - x$ .

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Step 6.4.4.1.2.1

Subtract $x$ from $x$ .

$0 + 3 = 0$

Step 6.4.4.1.2.2

Add $0$ and $3$ .

$3 = 0$

Step 6.4.4.2

Since $3 \neq 0$ , there are no solutions.

No solution

Step 6.4.5

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

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Step 6.4.5.1

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

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Step 6.4.5.1.1

Add $x$ to both sides of the equation.

$x + 3 + x = 0$

Step 6.4.5.1.2

Add $x$ and $x$ .

$2 x + 3 = 0$

Step 6.4.5.2

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 6.4.5.3

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

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Step 6.4.5.3.1

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

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

Step 6.4.5.3.2

Simplify the left side.

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Step 6.4.5.3.2.1

Cancel the common factor of $2$ .

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Step 6.4.5.3.2.1.1

Cancel the common factor.

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

Step 6.4.5.3.2.1.2

Divide $x$ by $1$ .

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

Step 6.4.5.3.3

Simplify the right side.

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Step 6.4.5.3.3.1

Move the negative in front of the fraction.

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

Step 6.4.6

Consolidate the solutions.

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

Step 6.5

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

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Step 6.5.1

Solve for $| x + 3 |$ .

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Step 6.5.1.1

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

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

Step 6.5.1.2

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

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Step 6.5.1.2.1

Apply the distributive property.

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

Step 6.5.1.2.2

Multiply $3$ by $- 1$ .

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

Step 6.5.1.3

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

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Step 6.5.1.3.1

Add $3 x$ to both sides of the equation.

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

Step 6.5.1.3.2

Add $4 x$ and $3 x$ .

$- | x + 3 | = 7 x$

Step 6.5.1.4

Divide each term in $- | x + 3 | = 7 x$ by $- 1$ and simplify.

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Step 6.5.1.4.1

Divide each term in $- | x + 3 | = 7 x$ by $- 1$ .

$\frac{- | x + 3 |}{- 1} = \frac{7 x}{- 1}$

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.

$\frac{| x + 3 |}{1} = \frac{7 x}{- 1}$

Step 6.5.1.4.2.2

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

$| x + 3 | = \frac{7 x}{- 1}$

Step 6.5.1.4.3

Simplify the right side.

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Step 6.5.1.4.3.1

Move the negative one from the denominator of $\frac{7 x}{- 1}$ .

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

Step 6.5.1.4.3.2

Rewrite $- 1 \cdot (7 x)$ as $- (7 x)$ .

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

Step 6.5.1.4.3.3

Multiply $7$ by $- 1$ .

$| x + 3 | = - 7 x$

Step 6.5.2

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

$x + 3 = \pm (- 7 x)$

Step 6.5.3

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

$x + 3 = - 7 x$

$x + 3 = - (- 7 x)$

Step 6.5.4

Solve $x + 3 = - 7 x$ for $x$ .

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Step 6.5.4.1

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

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Step 6.5.4.1.1

Add $7 x$ to both sides of the equation.

$x + 3 + 7 x = 0$

Step 6.5.4.1.2

Add $x$ and $7 x$ .

$8 x + 3 = 0$

Step 6.5.4.2

Subtract $3$ from both sides of the equation.

$8 x = - 3$

Step 6.5.4.3

Divide each term in $8 x = - 3$ by $8$ and simplify.

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Step 6.5.4.3.1

Divide each term in $8 x = - 3$ by $8$ .

$\frac{8 x}{8} = \frac{- 3}{8}$

Step 6.5.4.3.2

Simplify the left side.

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Step 6.5.4.3.2.1

Cancel the common factor of $8$ .

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Step 6.5.4.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 3}{8}$

Step 6.5.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{8}$

Step 6.5.4.3.3

Simplify the right side.

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Step 6.5.4.3.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{8}$

Step 6.5.5

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

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Step 6.5.5.1

Multiply $- 7$ by $- 1$ .

$x + 3 = 7 x$

Step 6.5.5.2

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

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Step 6.5.5.2.1

Subtract $7 x$ from both sides of the equation.

$x + 3 - 7 x = 0$

Step 6.5.5.2.2

Subtract $7 x$ from $x$ .

$- 6 x + 3 = 0$

Step 6.5.5.3

Subtract $3$ from both sides of the equation.

$- 6 x = - 3$

Step 6.5.5.4

Divide each term in $- 6 x = - 3$ by $- 6$ and simplify.

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Step 6.5.5.4.1

Divide each term in $- 6 x = - 3$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 3}{- 6}$

Step 6.5.5.4.2

Simplify the left side.

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Step 6.5.5.4.2.1

Cancel the common factor of $- 6$ .

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Step 6.5.5.4.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 3}{- 6}$

Step 6.5.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 6}$

Step 6.5.5.4.3

Simplify the right side.

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Step 6.5.5.4.3.1

Cancel the common factor of $- 3$ and $- 6$ .

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Step 6.5.5.4.3.1.1

Factor $- 3$ out of $- 3$ .

$x = \frac{- 3 (1)}{- 6}$

Step 6.5.5.4.3.1.2

Cancel the common factors.

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Step 6.5.5.4.3.1.2.1

Factor $- 3$ out of $- 6$ .

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

Step 6.5.5.4.3.1.2.2

Cancel the common factor.

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

Step 6.5.5.4.3.1.2.3

Rewrite the expression.

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

Step 6.5.6

Consolidate the solutions.

$x = - \frac{3}{8}, \frac{1}{2}$

Step 6.6

Consolidate the solutions.

$x = - \frac{3}{2}, - \frac{3}{8}, \frac{1}{2}$

Step 7

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

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Step 7.1

Solve for $| 4 x |$ .

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Step 7.1.1

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

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

Step 7.1.2

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

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Step 7.1.2.1

Apply the distributive property.

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

Step 7.1.2.2

Multiply $- - | x + 3 |$ .

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Step 7.1.2.2.1

Multiply $- 1$ by $- 1$ .

$1 | x + 3 | - | 4 x | = 3 x$

Step 7.1.2.2.2

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

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

Step 7.1.3

Subtract $| x + 3 |$ from both sides of the equation.

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

Step 7.1.4

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

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Step 7.1.4.1

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

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

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.

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

Step 7.1.4.2.2

Divide $| 4 x |$ by $1$ .

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

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 $\frac{3 x}{- 1}$ .

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

Step 7.1.4.3.1.2

Rewrite $- 1 \cdot (3 x)$ as $- (3 x)$ .

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

Step 7.1.4.3.1.3

Multiply $3$ by $- 1$ .

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

Step 7.1.4.3.1.4

Dividing two negative values results in a positive value.

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

Step 7.1.4.3.1.5

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

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

Step 7.2

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

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

Step 7.3

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

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

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

Step 7.4

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

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Step 7.4.1

Solve for $| x + 3 |$ .

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Step 7.4.1.1

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

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

Step 7.4.1.2

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

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Step 7.4.1.2.1

Add $3 x$ to both sides of the equation.

$| x + 3 | = 4 x + 3 x$

Step 7.4.1.2.2

Add $4 x$ and $3 x$ .

$| x + 3 | = 7 x$

Step 7.4.2

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

$x + 3 = \pm (7 x)$

Step 7.4.3

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

$x + 3 = 7 x$

$x + 3 = - (7 x)$

Step 7.4.4

Solve $x + 3 = 7 x$ for $x$ .

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Step 7.4.4.1

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

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Step 7.4.4.1.1

Subtract $7 x$ from both sides of the equation.

$x + 3 - 7 x = 0$

Step 7.4.4.1.2

Subtract $7 x$ from $x$ .

$- 6 x + 3 = 0$

Step 7.4.4.2

Subtract $3$ from both sides of the equation.

$- 6 x = - 3$

Step 7.4.4.3

Divide each term in $- 6 x = - 3$ by $- 6$ and simplify.

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Step 7.4.4.3.1

Divide each term in $- 6 x = - 3$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 3}{- 6}$

Step 7.4.4.3.2

Simplify the left side.

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Step 7.4.4.3.2.1

Cancel the common factor of $- 6$ .

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Step 7.4.4.3.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 3}{- 6}$

Step 7.4.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 6}$

Step 7.4.4.3.3

Simplify the right side.

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Step 7.4.4.3.3.1

Cancel the common factor of $- 3$ and $- 6$ .

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Step 7.4.4.3.3.1.1

Factor $- 3$ out of $- 3$ .

$x = \frac{- 3 (1)}{- 6}$

Step 7.4.4.3.3.1.2

Cancel the common factors.

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Step 7.4.4.3.3.1.2.1

Factor $- 3$ out of $- 6$ .

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

Step 7.4.4.3.3.1.2.2

Cancel the common factor.

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

Step 7.4.4.3.3.1.2.3

Rewrite the expression.

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

Step 7.4.5

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

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Step 7.4.5.1

Multiply $7$ by $- 1$ .

$x + 3 = - 7 x$

Step 7.4.5.2

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

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Step 7.4.5.2.1

Add $7 x$ to both sides of the equation.

$x + 3 + 7 x = 0$

Step 7.4.5.2.2

Add $x$ and $7 x$ .

$8 x + 3 = 0$

Step 7.4.5.3

Subtract $3$ from both sides of the equation.

$8 x = - 3$

Step 7.4.5.4

Divide each term in $8 x = - 3$ by $8$ and simplify.

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Step 7.4.5.4.1

Divide each term in $8 x = - 3$ by $8$ .

$\frac{8 x}{8} = \frac{- 3}{8}$

Step 7.4.5.4.2

Simplify the left side.

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Step 7.4.5.4.2.1

Cancel the common factor of $8$ .

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Step 7.4.5.4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 3}{8}$

Step 7.4.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{8}$

Step 7.4.5.4.3

Simplify the right side.

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Step 7.4.5.4.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{8}$

Step 7.4.6

Consolidate the solutions.

$x = \frac{1}{2}, - \frac{3}{8}$

Step 7.5

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

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Step 7.5.1

Solve for $| x + 3 |$ .

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Step 7.5.1.1

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

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

Step 7.5.1.2

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

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Step 7.5.1.2.1

Apply the distributive property.

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

Step 7.5.1.2.2

Multiply $- 3$ by $- 1$ .

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

Step 7.5.1.3

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

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Step 7.5.1.3.1

Subtract $3 x$ from both sides of the equation.

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

Step 7.5.1.3.2

Subtract $3 x$ from $4 x$ .

$- | x + 3 | = x$

Step 7.5.1.4

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

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Step 7.5.1.4.1

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

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

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.

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

Step 7.5.1.4.2.2

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

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

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

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

Step 7.5.1.4.3.2

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

$| x + 3 | = - x$

Step 7.5.2

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

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

Step 7.5.3

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

$x + 3 = - x$

$x + 3 = - (- x)$

Step 7.5.4

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

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Step 7.5.4.1

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

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Step 7.5.4.1.1

Add $x$ to both sides of the equation.

$x + 3 + x = 0$

Step 7.5.4.1.2

Add $x$ and $x$ .

$2 x + 3 = 0$

Step 7.5.4.2

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 7.5.4.3

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

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Step 7.5.4.3.1

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

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

Step 7.5.4.3.2

Simplify the left side.

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Step 7.5.4.3.2.1

Cancel the common factor of $2$ .

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Step 7.5.4.3.2.1.1

Cancel the common factor.

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

Step 7.5.4.3.2.1.2

Divide $x$ by $1$ .

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

Step 7.5.4.3.3

Simplify the right side.

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Step 7.5.4.3.3.1

Move the negative in front of the fraction.

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

Step 7.5.5

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

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Step 7.5.5.1

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

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Step 7.5.5.1.1

Subtract $x$ from both sides of the equation.

$x + 3 - x = 0$

Step 7.5.5.1.2

Combine the opposite terms in $x + 3 - x$ .

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Step 7.5.5.1.2.1

Subtract $x$ from $x$ .

$0 + 3 = 0$

Step 7.5.5.1.2.2

Add $0$ and $3$ .

$3 = 0$

Step 7.5.5.2

Since $3 \neq 0$ , there are no solutions.

No solution

Step 7.5.6

Consolidate the solutions.

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

Step 7.6

Consolidate the solutions.

$x = \frac{1}{2}, - \frac{3}{8}, - \frac{3}{2}$

Step 8

Consolidate the solutions.

$x = - \frac{3}{2}, - \frac{3}{8}, \frac{1}{2}$

Step 9

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

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1.5$

Mixed Number Form:

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