Trigonometry Examples

$6 - 3 y > 9 x$

Step 1

Solve for $y$ .

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Step 1.1

Subtract $6$ from both sides of the inequality.

$- 3 y > 9 x - 6$

Step 1.2

Divide each term in $- 3 y > 9 x - 6$ by $- 3$ and simplify.

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Step 1.2.1

Divide each term in $- 3 y > 9 x - 6$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 y}{- 3} < \frac{9 x}{- 3} + \frac{- 6}{- 3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} < \frac{9 x}{- 3} + \frac{- 6}{- 3}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y < \frac{9 x}{- 3} + \frac{- 6}{- 3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $9$ and $- 3$ .

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Step 1.2.3.1.1.1

Factor $3$ out of $9 x$ .

$y < \frac{3 (3 x)}{- 3} + \frac{- 6}{- 3}$

Step 1.2.3.1.1.2

Move the negative one from the denominator of $\frac{3 x}{- 1}$ .

$y < - 1 \cdot (3 x) + \frac{- 6}{- 3}$

Step 1.2.3.1.2

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

$y < - (3 x) + \frac{- 6}{- 3}$

Step 1.2.3.1.3

Multiply $3$ by $- 1$ .

$y < - 3 x + \frac{- 6}{- 3}$

Step 1.2.3.1.4

Divide $- 6$ by $- 3$ .

$y < - 3 x + 2$

Step 2

Use the slope-intercept form to find the slope and y-intercept.

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Step 2.1

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 2.2

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = - 3$

$b = 2$

Step 2.3

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $- 3$

y-intercept: $(0, 2)$

Slope: $- 3$

y-intercept: $(0, 2)$

Step 3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $- 3 x + 2$ .

$y < - 3 x + 2$

Step 4