Finite Math Examples

$| 5 x - 9 | = 2 (x + 4)$

Step 1

Simplify $2 (x + 4)$ .

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Step 1.1

Apply the distributive property.

$| 5 x - 9 | = 2 x + 2 \cdot 4$

Step 1.2

Multiply $2$ by $4$ .

$| 5 x - 9 | = 2 x + 8$

Step 2

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

$5 x - 9 = \pm (2 x + 8)$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$5 x - 9 = 2 x + 8$

Step 3.2

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

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Step 3.2.1

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

$5 x - 9 - 2 x = 8$

Step 3.2.2

Subtract $2 x$ from $5 x$ .

$3 x - 9 = 8$

Step 3.3

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

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Step 3.3.1

Add $9$ to both sides of the equation.

$3 x = 8 + 9$

Step 3.3.2

Add $8$ and $9$ .

$3 x = 17$

Step 3.4

Divide each term in $3 x = 17$ by $3$ and simplify.

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Step 3.4.1

Divide each term in $3 x = 17$ by $3$ .

$\frac{3 x}{3} = \frac{17}{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{17}{3}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{17}{3}$

Step 3.5

Next, use the negative value of the $\pm$ to find the second solution.

$5 x - 9 = - (2 x + 8)$

Step 3.6

Simplify $- (2 x + 8)$ .

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Step 3.6.1

Rewrite.

$5 x - 9 = 0 + 0 - (2 x + 8)$

Step 3.6.2

Simplify by adding zeros.

$5 x - 9 = - (2 x + 8)$

Step 3.6.3

Apply the distributive property.

$5 x - 9 = - (2 x) - 1 \cdot 8$

Step 3.6.4

Multiply.

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Step 3.6.4.1

Multiply $2$ by $- 1$ .

$5 x - 9 = - 2 x - 1 \cdot 8$

Step 3.6.4.2

Multiply $- 1$ by $8$ .

$5 x - 9 = - 2 x - 8$

Step 3.7

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

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Step 3.7.1

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

$5 x - 9 + 2 x = - 8$

Step 3.7.2

Add $5 x$ and $2 x$ .

$7 x - 9 = - 8$

Step 3.8

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

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Step 3.8.1

Add $9$ to both sides of the equation.

$7 x = - 8 + 9$

Step 3.8.2

Add $- 8$ and $9$ .

$7 x = 1$

Step 3.9

Divide each term in $7 x = 1$ by $7$ and simplify.

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Step 3.9.1

Divide each term in $7 x = 1$ by $7$ .

$\frac{7 x}{7} = \frac{1}{7}$

Step 3.9.2

Simplify the left side.

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Step 3.9.2.1

Cancel the common factor of $7$ .

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Step 3.9.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{1}{7}$

Step 3.9.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{7}$

Step 3.10

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{17}{3}, \frac{1}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{17}{3}, \frac{1}{7}$

Decimal Form:

$x = 5.\overline{6}, 0.\overline{142857}$

Mixed Number Form:

$x = 5 \frac{2}{3}, \frac{1}{7}$