Precalculus Examples

$| \frac{5 x + 2}{x - 1} | = 3$

Step 1

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

$\frac{5 x + 2}{x - 1} = \pm 3$

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.

$\frac{5 x + 2}{x - 1} = 3$

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.

$x - 1, 1$

Step 2.2.2

Remove parentheses.

$x - 1, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$x - 1$

Step 2.3

Multiply each term in $\frac{5 x + 2}{x - 1} = 3$ by $x - 1$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{5 x + 2}{x - 1} = 3$ by $x - 1$ .

$\frac{5 x + 2}{x - 1} (x - 1) = 3 (x - 1)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $x - 1$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{5 x + 2}{x - 1} (x - 1) = 3 (x - 1)$

Step 2.3.2.1.2

Rewrite the expression.

$5 x + 2 = 3 (x - 1)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$5 x + 2 = 3 x + 3 \cdot - 1$

Step 2.3.3.2

Multiply $3$ by $- 1$ .

$5 x + 2 = 3 x - 3$

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 $3 x$ from both sides of the equation.

$5 x + 2 - 3 x = - 3$

Step 2.4.1.2

Subtract $3 x$ from $5 x$ .

$2 x + 2 = - 3$

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.

$2 x = - 3 - 2$

Step 2.4.2.2

Subtract $2$ from $- 3$ .

$2 x = - 5$

Step 2.4.3

Divide each term in $2 x = - 5$ by $2$ and simplify.

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Step 2.4.3.1

Divide each term in $2 x = - 5$ by $2$ .

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

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $2$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

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

Step 2.4.3.2.1.2

Divide $x$ by $1$ .

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

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.

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

Step 2.5

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

$\frac{5 x + 2}{x - 1} = - 3$

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.

$x - 1, 1$

Step 2.6.2

Remove parentheses.

$x - 1, 1$

Step 2.6.3

The LCM of one and any expression is the expression.

$x - 1$

Step 2.7

Multiply each term in $\frac{5 x + 2}{x - 1} = - 3$ by $x - 1$ to eliminate the fractions.

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Step 2.7.1

Multiply each term in $\frac{5 x + 2}{x - 1} = - 3$ by $x - 1$ .

$\frac{5 x + 2}{x - 1} (x - 1) = - 3 (x - 1)$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Cancel the common factor of $x - 1$ .

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Step 2.7.2.1.1

Cancel the common factor.

$\frac{5 x + 2}{x - 1} (x - 1) = - 3 (x - 1)$

Step 2.7.2.1.2

Rewrite the expression.

$5 x + 2 = - 3 (x - 1)$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

Apply the distributive property.

$5 x + 2 = - 3 x - 3 \cdot - 1$

Step 2.7.3.2

Multiply $- 3$ by $- 1$ .

$5 x + 2 = - 3 x + 3$

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 $3 x$ to both sides of the equation.

$5 x + 2 + 3 x = 3$

Step 2.8.1.2

Add $5 x$ and $3 x$ .

$8 x + 2 = 3$

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.

$8 x = 3 - 2$

Step 2.8.2.2

Subtract $2$ from $3$ .

$8 x = 1$

Step 2.8.3

Divide each term in $8 x = 1$ by $8$ and simplify.

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Step 2.8.3.1

Divide each term in $8 x = 1$ by $8$ .

$\frac{8 x}{8} = \frac{1}{8}$

Step 2.8.3.2

Simplify the left side.

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Step 2.8.3.2.1

Cancel the common factor of $8$ .

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Step 2.8.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{1}{8}$

Step 2.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{8}$

Step 2.9

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

$x = - \frac{5}{2}, \frac{1}{8}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{2}, \frac{1}{8}$

Decimal Form:

$x = - 2.5, 0.125$

Mixed Number Form:

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