Trigonometry Examples

$f (x) < (3 x^{2} - 24 x) - 3$

Step 1

Subtract $f (x)$ from both sides of the inequality.

$0 < 3 x^{2} - 24 x - 3 - f (x)$

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

Rewrite so $x$ is on the left side of the inequality.

$3 x^{2} - 24 x - 3 - f (x) > 0$

Step 2.1.3

Convert the inequality to an equation.

$3 x^{2} - 24 x - 3 - f (x) = 0$

Step 2.1.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.1.5

Substitute the values $a = 3$ , $b = - 24$ , and $c = - 3 - f (x)$ into the quadratic formula and solve for $x$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (3 \cdot (- 3 - f (x)))}}{2 \cdot 3}$

Step 2.1.6

Simplify.

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Step 2.1.6.1

Simplify the numerator.

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Step 2.1.6.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot (- 3 - f (x))}}{2 \cdot 3}$

Step 2.1.6.1.2

Multiply $- 4$ by $3$ .

$x = \frac{24 \pm \sqrt{576 - 12 \cdot (- 3 - f (x))}}{2 \cdot 3}$

Step 2.1.6.1.3

Apply the distributive property.

$x = \frac{24 \pm \sqrt{576 - 12 \cdot - 3 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.6.1.4

Multiply $- 12$ by $- 3$ .

$x = \frac{24 \pm \sqrt{576 + 36 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.6.1.5

Multiply $- 1$ by $- 12$ .

$x = \frac{24 \pm \sqrt{576 + 36 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.6.1.6

Add $576$ and $36$ .

$x = \frac{24 \pm \sqrt{612 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.6.2

Multiply $2$ by $3$ .

$x = \frac{24 \pm \sqrt{612 + 12 f (x)}}{6}$

Step 2.1.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.1.7.1

Simplify the numerator.

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Step 2.1.7.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot (- 3 - f (x))}}{2 \cdot 3}$

Step 2.1.7.1.2

Multiply $- 4$ by $3$ .

$x = \frac{24 \pm \sqrt{576 - 12 \cdot (- 3 - f (x))}}{2 \cdot 3}$

Step 2.1.7.1.3

Apply the distributive property.

$x = \frac{24 \pm \sqrt{576 - 12 \cdot - 3 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.7.1.4

Multiply $- 12$ by $- 3$ .

$x = \frac{24 \pm \sqrt{576 + 36 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.7.1.5

Multiply $- 1$ by $- 12$ .

$x = \frac{24 \pm \sqrt{576 + 36 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.7.1.6

Add $576$ and $36$ .

$x = \frac{24 \pm \sqrt{612 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.7.2

Multiply $2$ by $3$ .

$x = \frac{24 \pm \sqrt{612 + 12 f (x)}}{6}$

Step 2.1.7.3

Change the $\pm$ to $+$ .

$x = \frac{24 + \sqrt{612 + 12 f (x)}}{6}$

Step 2.1.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.1.8.1

Simplify the numerator.

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Step 2.1.8.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 3 \cdot (- 3 - f (x))}}{2 \cdot 3}$

Step 2.1.8.1.2

Multiply $- 4$ by $3$ .

$x = \frac{24 \pm \sqrt{576 - 12 \cdot (- 3 - f (x))}}{2 \cdot 3}$

Step 2.1.8.1.3

Apply the distributive property.

$x = \frac{24 \pm \sqrt{576 - 12 \cdot - 3 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.8.1.4

Multiply $- 12$ by $- 3$ .

$x = \frac{24 \pm \sqrt{576 + 36 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.8.1.5

Multiply $- 1$ by $- 12$ .

$x = \frac{24 \pm \sqrt{576 + 36 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.8.1.6

Add $576$ and $36$ .

$x = \frac{24 \pm \sqrt{612 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.8.2

Multiply $2$ by $3$ .

$x = \frac{24 \pm \sqrt{612 + 12 f (x)}}{6}$

Step 2.1.8.3

Change the $\pm$ to $-$ .

$x = \frac{24 - \sqrt{612 + 12 f (x)}}{6}$

Step 2.1.9

Consolidate the solutions.

$x = \frac{24 + \sqrt{612 + 12 f (x)}}{6}, \frac{24 - \sqrt{612 + 12 f (x)}}{6}$

Step 2.1.10

Arrange the polynomial to follow the $y = m x + b$ form for slope and y-intercept.

$3 x^{2} - 24 x = f (x) + 3$

Step 2.2

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

Not Linear

Step 3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $3 x^{2} - 24 x - 3 - f (x)$ .

$0 < 3 x^{2} - 24 x - 3 - f (x)$

Step 4