Trigonometry Examples

$(x + 6) (x - 7) < o$

Step 1

Solve for $y$ .

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Step 1.1

Simplify $(x + 6) (x - 7)$ .

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Step 1.1.1

Expand $(x + 6) (x - 7)$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$x (x - 7) + 6 (x - 7) < o$

Step 1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 7 + 6 (x - 7) < o$

Step 1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 7 + 6 x + 6 \cdot - 7 < o$

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 7 + 6 x + 6 \cdot - 7 < o$

Step 1.1.2.1.2

Move $- 7$ to the left of $x$ .

$x^{2} - 7 \cdot x + 6 x + 6 \cdot - 7 < o$

Step 1.1.2.1.3

Multiply $6$ by $- 7$ .

$x^{2} - 7 x + 6 x - 42 < o$

Step 1.1.2.2

Add $- 7 x$ and $6 x$ .

$x^{2} - x - 42 < o$

Step 1.2

Subtract $o$ from both sides of the inequality.

$x^{2} - x - 42 - o < 0$

Step 1.3

Convert the inequality to an equation.

$x^{2} - x - 42 - o = 0$

Step 1.4

Use the quadratic formula to find the solutions.

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

Step 1.5

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 42 - o$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot (- 42 - o))}}{2 \cdot 1}$

Step 1.6

Simplify.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot (- 42 - o)}}{2 \cdot 1}$

Step 1.6.1.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot (- 42 - o)}}{2 \cdot 1}$

Step 1.6.1.3

Apply the distributive property.

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 42 - 4 (- o)}}{2 \cdot 1}$

Step 1.6.1.4

Multiply $- 4$ by $- 42$ .

$x = \frac{1 \pm \sqrt{1 + 168 - 4 (- o)}}{2 \cdot 1}$

Step 1.6.1.5

Multiply $- 1$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 + 168 + 4 o}}{2 \cdot 1}$

Step 1.6.1.6

Add $1$ and $168$ .

$x = \frac{1 \pm \sqrt{169 + 4 o}}{2 \cdot 1}$

Step 1.6.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{169 + 4 o}}{2}$

Step 1.7

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

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Step 1.7.1

Simplify the numerator.

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Step 1.7.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot (- 42 - o)}}{2 \cdot 1}$

Step 1.7.1.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot (- 42 - o)}}{2 \cdot 1}$

Step 1.7.1.3

Apply the distributive property.

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 42 - 4 (- o)}}{2 \cdot 1}$

Step 1.7.1.4

Multiply $- 4$ by $- 42$ .

$x = \frac{1 \pm \sqrt{1 + 168 - 4 (- o)}}{2 \cdot 1}$

Step 1.7.1.5

Multiply $- 1$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 + 168 + 4 o}}{2 \cdot 1}$

Step 1.7.1.6

Add $1$ and $168$ .

$x = \frac{1 \pm \sqrt{169 + 4 o}}{2 \cdot 1}$

Step 1.7.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{169 + 4 o}}{2}$

Step 1.7.3

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{169 + 4 o}}{2}$

Step 1.8

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

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Step 1.8.1

Simplify the numerator.

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Step 1.8.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot (- 42 - o)}}{2 \cdot 1}$

Step 1.8.1.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot (- 42 - o)}}{2 \cdot 1}$

Step 1.8.1.3

Apply the distributive property.

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 42 - 4 (- o)}}{2 \cdot 1}$

Step 1.8.1.4

Multiply $- 4$ by $- 42$ .

$x = \frac{1 \pm \sqrt{1 + 168 - 4 (- o)}}{2 \cdot 1}$

Step 1.8.1.5

Multiply $- 1$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 + 168 + 4 o}}{2 \cdot 1}$

Step 1.8.1.6

Add $1$ and $168$ .

$x = \frac{1 \pm \sqrt{169 + 4 o}}{2 \cdot 1}$

Step 1.8.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{169 + 4 o}}{2}$

Step 1.8.3

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{169 + 4 o}}{2}$

Step 1.9

Consolidate the solutions.

$x = \frac{1 + \sqrt{169 + 4 o}}{2}$

$x = \frac{1 - \sqrt{169 + 4 o}}{2}$

$x = \frac{1 + \sqrt{169 + 4 o}}{2}$

$x = \frac{1 - \sqrt{169 + 4 o}}{2}$

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

Remove parentheses.

$x = \frac{1 + \sqrt{169 + 4 o}}{2}$

$x = \frac{1 - \sqrt{169 + 4 o}}{2}$

Step 2.1.3

Simplify $\frac{1 + \sqrt{169 + 4 o}}{2}, \frac{1 - \sqrt{169 + 4 o}}{2}$ .

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Step 2.1.3.1

Split the fraction $\frac{1 + \sqrt{169 + 4 o}}{2}$ into two fractions.

$x = \frac{1}{2} + \frac{\sqrt{169 + 4 o}}{2}$

$x = \frac{1 - \sqrt{169 + 4 o}}{2}$

Step 2.1.3.2

Split the fraction $\frac{1 - \sqrt{169 + 4 o}}{2}$ into two fractions.

$x = \frac{1}{2} + \frac{\sqrt{169 + 4 o}}{2}$

$x = \frac{1}{2} + \frac{- \sqrt{169 + 4 o}}{2}$

Step 2.1.3.3

Move the negative in front of the fraction.

$x = \frac{1}{2} + \frac{\sqrt{169 + 4 o}}{2}$

$x = \frac{1}{2} - \frac{\sqrt{169 + 4 o}}{2}$

$x = \frac{1}{2} + \frac{\sqrt{169 + 4 o}}{2}$

$x = \frac{1}{2} - \frac{\sqrt{169 + 4 o}}{2}$

Step 2.1.4

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

$x = \frac{\sqrt{169 + 4 o}}{2} + \frac{1}{2} + \frac{1}{2}$

Step 2.1.5

Combine $\frac{1}{2} + \frac{1}{2}$ .

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Step 2.1.5.1

Combine the numerators over the common denominator.

$x = \frac{\sqrt{169 + 4 o}}{2} + \frac{1 + 1}{2}$

Step 2.1.5.2

Simplify the expression.

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Step 2.1.5.2.1

Add $1$ and $1$ .

$x = \frac{\sqrt{169 + 4 o}}{2} + \frac{2}{2}$

Step 2.1.5.2.2

Divide $2$ by $2$ .

$x = \frac{\sqrt{169 + 4 o}}{2} + 1$

Step 2.1.6

Rewrite in slope-intercept form.

$= \frac{\sqrt{169 + 4}}{2} o + 1$

Step 2.2

Since the equation is a vertical line, it does not cross the y-axis.

No y-intercept

Step 2.3

Since the equation is a vertical line, the slope is infinite.

$m = \infty$

Step 3