Precalculus Examples

$\frac{9}{2 \cdot (x + 1)} + \frac{1}{x - 1} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1

Simplify $\frac{9}{2 \cdot (x + 1)} + \frac{1}{x - 1}$ .

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Step 1.1

To write $\frac{9}{2 (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{9}{2 (x + 1)} \cdot \frac{x - 1}{x - 1} + \frac{1}{x - 1} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.2

To write $\frac{1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{2 (x + 1)}{2 (x + 1)}$ .

$\frac{9}{2 (x + 1)} \cdot \frac{x - 1}{x - 1} + \frac{1}{x - 1} \cdot \frac{2 (x + 1)}{2 (x + 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.3

Write each expression with a common denominator of $2 (x + 1) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{9}{2 (x + 1)}$ by $\frac{x - 1}{x - 1}$ .

$\frac{9 (x - 1)}{2 (x + 1) (x - 1)} + \frac{1}{x - 1} \cdot \frac{2 (x + 1)}{2 (x + 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.3.2

Multiply $\frac{1}{x - 1}$ by $\frac{2 (x + 1)}{2 (x + 1)}$ .

$\frac{9 (x - 1)}{2 (x + 1) (x - 1)} + \frac{2 (x + 1)}{(x - 1) (2 (x + 1))} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.3.3

Reorder the factors of $(x - 1) (2 (x + 1))$ .

$\frac{9 (x - 1)}{2 (x + 1) (x - 1)} + \frac{2 (x + 1)}{2 (x + 1) (x - 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{9 (x - 1) + 2 (x + 1)}{2 (x + 1) (x - 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{9 x + 9 \cdot - 1 + 2 (x + 1)}{2 (x + 1) (x - 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.5.2

Multiply $9$ by $- 1$ .

$\frac{9 x - 9 + 2 (x + 1)}{2 (x + 1) (x - 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.5.3

Apply the distributive property.

$\frac{9 x - 9 + 2 x + 2 \cdot 1}{2 (x + 1) (x - 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.5.4

Multiply $2$ by $1$ .

$\frac{9 x - 9 + 2 x + 2}{2 (x + 1) (x - 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.5.5

Add $9 x$ and $2 x$ .

$\frac{11 x - 9 + 2}{2 (x + 1) (x - 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 1.5.6

Add $- 9$ and $2$ .

$\frac{11 x - 7}{2 (x + 1) (x - 1)} = \frac{x + 1}{2 \cdot (2 - 1)}$

Step 2

Simplify $\frac{x + 1}{2 \cdot (2 - 1)}$ .

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Step 2.1

Subtract $1$ from $2$ .

$\frac{11 x - 7}{2 (x + 1) (x - 1)} = \frac{x + 1}{2 \cdot 1}$

Step 2.2

Multiply $2$ by $1$ .

$\frac{11 x - 7}{2 (x + 1) (x - 1)} = \frac{x + 1}{2}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 4.1985232, 0.53692278, 2.66160041$

Step 4