Algebra Examples

$y \leq \frac{1}{x - 2} - 1$

Step 1

Simplify $\frac{1}{x - 2} - 1$ .

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Step 1.1

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

$y \leq \frac{1}{x - 2} - 1 \cdot \frac{x - 2}{x - 2}$

Step 1.2

Simplify terms.

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Step 1.2.1

Combine $- 1$ and $\frac{x - 2}{x - 2}$ .

$y \leq \frac{1}{x - 2} + \frac{- (x - 2)}{x - 2}$

Step 1.2.2

Combine the numerators over the common denominator.

$y \leq \frac{1 - (x - 2)}{x - 2}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

Apply the distributive property.

$y \leq \frac{1 - x - - 2}{x - 2}$

Step 1.3.2

Multiply $- 1$ by $- 2$ .

$y \leq \frac{1 - x + 2}{x - 2}$

Step 1.3.3

Add $1$ and $2$ .

$y \leq \frac{- x + 3}{x - 2}$

Step 1.4

Simplify with factoring out.

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Step 1.4.1

Factor $- 1$ out of $- x$ .

$y \leq \frac{- (x) + 3}{x - 2}$

Step 1.4.2

Rewrite $3$ as $- 1 (- 3)$ .

$y \leq \frac{- (x) - 1 (- 3)}{x - 2}$

Step 1.4.3

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

$y \leq \frac{- (x - 3)}{x - 2}$

Step 1.4.4

Simplify the expression.

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Step 1.4.4.1

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$y \leq \frac{- 1 (x - 3)}{x - 2}$

Step 1.4.4.2

Move the negative in front of the fraction.

$y \leq - \frac{x - 3}{x - 2}$

Step 2

Find the slope and the y-intercept for the boundary line.

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.1.2

Simplify $- \frac{x - 3}{x - 2}$ .

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Step 2.1.2.1

Split the fraction $\frac{x - 3}{x - 2}$ into two fractions.

$y \leq - (\frac{x}{x - 2} + \frac{- 3}{x - 2})$

Step 2.1.2.2

Move the negative in front of the fraction.

$y \leq - (\frac{x}{x - 2} - \frac{3}{x - 2})$

Step 2.1.2.3

Apply the distributive property.

$y \leq - \frac{x}{x - 2} - - \frac{3}{x - 2}$

Step 2.1.2.4

Multiply $- - \frac{3}{x - 2}$ .

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Step 2.1.2.4.1

Multiply $- 1$ by $- 1$ .

$y \leq - \frac{x}{x - 2} + 1 \frac{3}{x - 2}$

Step 2.1.2.4.2

Multiply $\frac{3}{x - 2}$ by $1$ .

$y \leq - \frac{x}{x - 2} + \frac{3}{x - 2}$

Step 2.1.3

Rewrite in slope-intercept form.

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

Step 2.2

The equation is not linear, so a constant slope does not exist.

Not Linear

Step 3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $- \frac{x - 3}{x - 2}$ .

$y \leq - \frac{x - 3}{x - 2}$

Step 4