Finite Math Examples

$| \frac{9 x - 6}{4} | = 10$

Step 1

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

$\frac{9 x - 6}{4} = \pm 10$

Step 2

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

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Step 2.1

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

$\frac{9 x - 6}{4} = 10$

Step 2.2

Multiply both sides by $4$ .

$\frac{9 x - 6}{4} \cdot 4 = 10 \cdot 4$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{9 x - 6}{4} \cdot 4$ .

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Step 2.3.1.1.1

Factor $3$ out of $9 x - 6$ .

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Step 2.3.1.1.1.1

Factor $3$ out of $9 x$ .

$\frac{3 (3 x) - 6}{4} \cdot 4 = 10 \cdot 4$

Step 2.3.1.1.1.2

Factor $3$ out of $- 6$ .

$\frac{3 (3 x) + 3 (- 2)}{4} \cdot 4 = 10 \cdot 4$

Step 2.3.1.1.1.3

Factor $3$ out of $3 (3 x) + 3 (- 2)$ .

$\frac{3 (3 x - 2)}{4} \cdot 4 = 10 \cdot 4$

Step 2.3.1.1.2

Cancel the common factor of $4$ .

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Step 2.3.1.1.2.1

Cancel the common factor.

$\frac{3 (3 x - 2)}{4} \cdot 4 = 10 \cdot 4$

Step 2.3.1.1.2.2

Rewrite the expression.

$3 (3 x - 2) = 10 \cdot 4$

Step 2.3.1.1.3

Apply the distributive property.

$3 (3 x) + 3 \cdot - 2 = 10 \cdot 4$

Step 2.3.1.1.4

Multiply.

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Step 2.3.1.1.4.1

Multiply $3$ by $3$ .

$9 x + 3 \cdot - 2 = 10 \cdot 4$

Step 2.3.1.1.4.2

Multiply $3$ by $- 2$ .

$9 x - 6 = 10 \cdot 4$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $10$ by $4$ .

$9 x - 6 = 40$

Step 2.4

Solve for $x$ .

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Step 2.4.1

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

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Step 2.4.1.1

Add $6$ to both sides of the equation.

$9 x = 40 + 6$

Step 2.4.1.2

Add $40$ and $6$ .

$9 x = 46$

Step 2.4.2

Divide each term in $9 x = 46$ by $9$ and simplify.

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Step 2.4.2.1

Divide each term in $9 x = 46$ by $9$ .

$\frac{9 x}{9} = \frac{46}{9}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $9$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{46}{9}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{46}{9}$

Step 2.5

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

$\frac{9 x - 6}{4} = - 10$

Step 2.6

Multiply both sides by $4$ .

$\frac{9 x - 6}{4} \cdot 4 = - 10 \cdot 4$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the left side.

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Step 2.7.1.1

Simplify $\frac{9 x - 6}{4} \cdot 4$ .

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Step 2.7.1.1.1

Factor $3$ out of $9 x - 6$ .

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Step 2.7.1.1.1.1

Factor $3$ out of $9 x$ .

$\frac{3 (3 x) - 6}{4} \cdot 4 = - 10 \cdot 4$

Step 2.7.1.1.1.2

Factor $3$ out of $- 6$ .

$\frac{3 (3 x) + 3 (- 2)}{4} \cdot 4 = - 10 \cdot 4$

Step 2.7.1.1.1.3

Factor $3$ out of $3 (3 x) + 3 (- 2)$ .

$\frac{3 (3 x - 2)}{4} \cdot 4 = - 10 \cdot 4$

Step 2.7.1.1.2

Cancel the common factor of $4$ .

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Step 2.7.1.1.2.1

Cancel the common factor.

$\frac{3 (3 x - 2)}{4} \cdot 4 = - 10 \cdot 4$

Step 2.7.1.1.2.2

Rewrite the expression.

$3 (3 x - 2) = - 10 \cdot 4$

Step 2.7.1.1.3

Apply the distributive property.

$3 (3 x) + 3 \cdot - 2 = - 10 \cdot 4$

Step 2.7.1.1.4

Multiply.

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Step 2.7.1.1.4.1

Multiply $3$ by $3$ .

$9 x + 3 \cdot - 2 = - 10 \cdot 4$

Step 2.7.1.1.4.2

Multiply $3$ by $- 2$ .

$9 x - 6 = - 10 \cdot 4$

Step 2.7.2

Simplify the right side.

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Step 2.7.2.1

Multiply $- 10$ by $4$ .

$9 x - 6 = - 40$

Step 2.8

Solve for $x$ .

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Step 2.8.1

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

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Step 2.8.1.1

Add $6$ to both sides of the equation.

$9 x = - 40 + 6$

Step 2.8.1.2

Add $- 40$ and $6$ .

$9 x = - 34$

Step 2.8.2

Divide each term in $9 x = - 34$ by $9$ and simplify.

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Step 2.8.2.1

Divide each term in $9 x = - 34$ by $9$ .

$\frac{9 x}{9} = \frac{- 34}{9}$

Step 2.8.2.2

Simplify the left side.

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Step 2.8.2.2.1

Cancel the common factor of $9$ .

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Step 2.8.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{- 34}{9}$

Step 2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 34}{9}$

Step 2.8.2.3

Simplify the right side.

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Step 2.8.2.3.1

Move the negative in front of the fraction.

$x = - \frac{34}{9}$

Step 2.9

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

$x = \frac{46}{9}, - \frac{34}{9}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{46}{9}, - \frac{34}{9}$

Decimal Form:

$x = 5.\overline{1}, - 3.\overline{7}$

Mixed Number Form:

$x = 5 \frac{1}{9}, - 3 \frac{7}{9}$