Finite Math Examples

$| \frac{4}{3} x + 7 | = \frac{3}{5}$

Step 1

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

$\frac{4}{3} x + 7 = \pm \frac{3}{5}$

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.

$\frac{4}{3} x + 7 = \frac{3}{5}$

Step 2.2

Combine $\frac{4}{3}$ and $x$ .

$\frac{4 x}{3} + 7 = \frac{3}{5}$

Step 2.3

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

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Step 2.3.1

Subtract $7$ from both sides of the equation.

$\frac{4 x}{3} = \frac{3}{5} - 7$

Step 2.3.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4 x}{3} = \frac{3}{5} - 7 \cdot \frac{5}{5}$

Step 2.3.3

Combine $- 7$ and $\frac{5}{5}$ .

$\frac{4 x}{3} = \frac{3}{5} + \frac{- 7 \cdot 5}{5}$

Step 2.3.4

Combine the numerators over the common denominator.

$\frac{4 x}{3} = \frac{3 - 7 \cdot 5}{5}$

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $- 7$ by $5$ .

$\frac{4 x}{3} = \frac{3 - 35}{5}$

Step 2.3.5.2

Subtract $35$ from $3$ .

$\frac{4 x}{3} = \frac{- 32}{5}$

Step 2.3.6

Move the negative in front of the fraction.

$\frac{4 x}{3} = - \frac{32}{5}$

Step 2.4

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (- \frac{32}{5})$

Step 2.5

Simplify both sides of the equation.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Simplify $\frac{3}{4} \cdot \frac{4 x}{3}$ .

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Step 2.5.1.1.1

Cancel the common factor of $3$ .

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Step 2.5.1.1.1.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (- \frac{32}{5})$

Step 2.5.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 x) = \frac{3}{4} (- \frac{32}{5})$

Step 2.5.1.1.2

Cancel the common factor of $4$ .

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Step 2.5.1.1.2.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} (4 (x)) = \frac{3}{4} (- \frac{32}{5})$

Step 2.5.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 x) = \frac{3}{4} (- \frac{32}{5})$

Step 2.5.1.1.2.3

Rewrite the expression.

$x = \frac{3}{4} (- \frac{32}{5})$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $\frac{3}{4} (- \frac{32}{5})$ .

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Step 2.5.2.1.1

Cancel the common factor of $4$ .

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Step 2.5.2.1.1.1

Move the leading negative in $- \frac{32}{5}$ into the numerator.

$x = \frac{3}{4} \cdot \frac{- 32}{5}$

Step 2.5.2.1.1.2

Factor $4$ out of $- 32$ .

$x = \frac{3}{4} \cdot \frac{4 (- 8)}{5}$

Step 2.5.2.1.1.3

Cancel the common factor.

$x = \frac{3}{4} \cdot \frac{4 \cdot - 8}{5}$

Step 2.5.2.1.1.4

Rewrite the expression.

$x = 3 (\frac{- 8}{5})$

Step 2.5.2.1.2

Combine $3$ and $\frac{- 8}{5}$ .

$x = \frac{3 \cdot - 8}{5}$

Step 2.5.2.1.3

Simplify the expression.

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Step 2.5.2.1.3.1

Multiply $3$ by $- 8$ .

$x = \frac{- 24}{5}$

Step 2.5.2.1.3.2

Move the negative in front of the fraction.

$x = - \frac{24}{5}$

Step 2.6

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

$\frac{4}{3} x + 7 = - \frac{3}{5}$

Step 2.7

Combine $\frac{4}{3}$ and $x$ .

$\frac{4 x}{3} + 7 = - \frac{3}{5}$

Step 2.8

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

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Step 2.8.1

Subtract $7$ from both sides of the equation.

$\frac{4 x}{3} = - \frac{3}{5} - 7$

Step 2.8.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4 x}{3} = - \frac{3}{5} - 7 \cdot \frac{5}{5}$

Step 2.8.3

Combine $- 7$ and $\frac{5}{5}$ .

$\frac{4 x}{3} = - \frac{3}{5} + \frac{- 7 \cdot 5}{5}$

Step 2.8.4

Combine the numerators over the common denominator.

$\frac{4 x}{3} = \frac{- 3 - 7 \cdot 5}{5}$

Step 2.8.5

Simplify the numerator.

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Step 2.8.5.1

Multiply $- 7$ by $5$ .

$\frac{4 x}{3} = \frac{- 3 - 35}{5}$

Step 2.8.5.2

Subtract $35$ from $- 3$ .

$\frac{4 x}{3} = \frac{- 38}{5}$

Step 2.8.6

Move the negative in front of the fraction.

$\frac{4 x}{3} = - \frac{38}{5}$

Step 2.9

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (- \frac{38}{5})$

Step 2.10

Simplify both sides of the equation.

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Step 2.10.1

Simplify the left side.

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Step 2.10.1.1

Simplify $\frac{3}{4} \cdot \frac{4 x}{3}$ .

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Step 2.10.1.1.1

Cancel the common factor of $3$ .

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Step 2.10.1.1.1.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (- \frac{38}{5})$

Step 2.10.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 x) = \frac{3}{4} (- \frac{38}{5})$

Step 2.10.1.1.2

Cancel the common factor of $4$ .

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Step 2.10.1.1.2.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} (4 (x)) = \frac{3}{4} (- \frac{38}{5})$

Step 2.10.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 x) = \frac{3}{4} (- \frac{38}{5})$

Step 2.10.1.1.2.3

Rewrite the expression.

$x = \frac{3}{4} (- \frac{38}{5})$

Step 2.10.2

Simplify the right side.

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Step 2.10.2.1

Simplify $\frac{3}{4} (- \frac{38}{5})$ .

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Step 2.10.2.1.1

Cancel the common factor of $2$ .

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Step 2.10.2.1.1.1

Move the leading negative in $- \frac{38}{5}$ into the numerator.

$x = \frac{3}{4} \cdot \frac{- 38}{5}$

Step 2.10.2.1.1.2

Factor $2$ out of $4$ .

$x = \frac{3}{2 (2)} \cdot \frac{- 38}{5}$

Step 2.10.2.1.1.3

Factor $2$ out of $- 38$ .

$x = \frac{3}{2 \cdot 2} \cdot \frac{2 \cdot - 19}{5}$

Step 2.10.2.1.1.4

Cancel the common factor.

$x = \frac{3}{2 \cdot 2} \cdot \frac{2 \cdot - 19}{5}$

Step 2.10.2.1.1.5

Rewrite the expression.

$x = \frac{3}{2} \cdot \frac{- 19}{5}$

Step 2.10.2.1.2

Multiply $\frac{3}{2}$ by $\frac{- 19}{5}$ .

$x = \frac{3 \cdot - 19}{2 \cdot 5}$

Step 2.10.2.1.3

Simplify the expression.

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Step 2.10.2.1.3.1

Multiply $3$ by $- 19$ .

$x = \frac{- 57}{2 \cdot 5}$

Step 2.10.2.1.3.2

Multiply $2$ by $5$ .

$x = \frac{- 57}{10}$

Step 2.10.2.1.3.3

Move the negative in front of the fraction.

$x = - \frac{57}{10}$

Step 2.11

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

$x = - \frac{24}{5}, - \frac{57}{10}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{24}{5}, - \frac{57}{10}$

Decimal Form:

$x = - 4.8, - 5.7$

Mixed Number Form:

$x = - 4 \frac{4}{5}, - 5 \frac{7}{10}$