Finite Math Examples

$| \frac{3 x + 2}{8 x + 7} | = 5$

Step 1

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

$\frac{3 x + 2}{8 x + 7} = \pm 5$

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.

$\frac{3 x + 2}{8 x + 7} = 5$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$8 x + 7, 1$

Step 2.2.2

Remove parentheses.

$8 x + 7, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$8 x + 7$

Step 2.3

Multiply each term in $\frac{3 x + 2}{8 x + 7} = 5$ by $8 x + 7$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{3 x + 2}{8 x + 7} = 5$ by $8 x + 7$ .

$\frac{3 x + 2}{8 x + 7} (8 x + 7) = 5 (8 x + 7)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $8 x + 7$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 x + 2}{8 x + 7} (8 x + 7) = 5 (8 x + 7)$

Step 2.3.2.1.2

Rewrite the expression.

$3 x + 2 = 5 (8 x + 7)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$3 x + 2 = 5 (8 x) + 5 \cdot 7$

Step 2.3.3.2

Multiply.

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Step 2.3.3.2.1

Multiply $8$ by $5$ .

$3 x + 2 = 40 x + 5 \cdot 7$

Step 2.3.3.2.2

Multiply $5$ by $7$ .

$3 x + 2 = 40 x + 35$

Step 2.4

Solve the equation.

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Step 2.4.1

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

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Step 2.4.1.1

Subtract $40 x$ from both sides of the equation.

$3 x + 2 - 40 x = 35$

Step 2.4.1.2

Subtract $40 x$ from $3 x$ .

$- 37 x + 2 = 35$

Step 2.4.2

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

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Step 2.4.2.1

Subtract $2$ from both sides of the equation.

$- 37 x = 35 - 2$

Step 2.4.2.2

Subtract $2$ from $35$ .

$- 37 x = 33$

Step 2.4.3

Divide each term in $- 37 x = 33$ by $- 37$ and simplify.

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Step 2.4.3.1

Divide each term in $- 37 x = 33$ by $- 37$ .

$\frac{- 37 x}{- 37} = \frac{33}{- 37}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $- 37$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{- 37 x}{- 37} = \frac{33}{- 37}$

Step 2.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{33}{- 37}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Move the negative in front of the fraction.

$x = - \frac{33}{37}$

Step 2.5

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

$\frac{3 x + 2}{8 x + 7} = - 5$

Step 2.6

Find the LCD of the terms in the equation.

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Step 2.6.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$8 x + 7, 1$

Step 2.6.2

Remove parentheses.

$8 x + 7, 1$

Step 2.6.3

The LCM of one and any expression is the expression.

$8 x + 7$

Step 2.7

Multiply each term in $\frac{3 x + 2}{8 x + 7} = - 5$ by $8 x + 7$ to eliminate the fractions.

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Step 2.7.1

Multiply each term in $\frac{3 x + 2}{8 x + 7} = - 5$ by $8 x + 7$ .

$\frac{3 x + 2}{8 x + 7} (8 x + 7) = - 5 (8 x + 7)$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Cancel the common factor of $8 x + 7$ .

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Step 2.7.2.1.1

Cancel the common factor.

$\frac{3 x + 2}{8 x + 7} (8 x + 7) = - 5 (8 x + 7)$

Step 2.7.2.1.2

Rewrite the expression.

$3 x + 2 = - 5 (8 x + 7)$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

Apply the distributive property.

$3 x + 2 = - 5 (8 x) - 5 \cdot 7$

Step 2.7.3.2

Multiply.

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Step 2.7.3.2.1

Multiply $8$ by $- 5$ .

$3 x + 2 = - 40 x - 5 \cdot 7$

Step 2.7.3.2.2

Multiply $- 5$ by $7$ .

$3 x + 2 = - 40 x - 35$

Step 2.8

Solve the equation.

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Step 2.8.1

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

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Step 2.8.1.1

Add $40 x$ to both sides of the equation.

$3 x + 2 + 40 x = - 35$

Step 2.8.1.2

Add $3 x$ and $40 x$ .

$43 x + 2 = - 35$

Step 2.8.2

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

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Step 2.8.2.1

Subtract $2$ from both sides of the equation.

$43 x = - 35 - 2$

Step 2.8.2.2

Subtract $2$ from $- 35$ .

$43 x = - 37$

Step 2.8.3

Divide each term in $43 x = - 37$ by $43$ and simplify.

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Step 2.8.3.1

Divide each term in $43 x = - 37$ by $43$ .

$\frac{43 x}{43} = \frac{- 37}{43}$

Step 2.8.3.2

Simplify the left side.

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Step 2.8.3.2.1

Cancel the common factor of $43$ .

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Step 2.8.3.2.1.1

Cancel the common factor.

$\frac{43 x}{43} = \frac{- 37}{43}$

Step 2.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 37}{43}$

Step 2.8.3.3

Simplify the right side.

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Step 2.8.3.3.1

Move the negative in front of the fraction.

$x = - \frac{37}{43}$

Step 2.9

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

$x = - \frac{33}{37}, - \frac{37}{43}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{33}{37}, - \frac{37}{43}$

Decimal Form:

$x = - 0.\overline{891}, - 0.86046511 \dots$