Algebra Examples

$f (x) < - x^{2} + 12 x - 29$

Step 1

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

$0 < - x^{2} + 12 x - 29 - f (x)$

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.

$- x^{2} + 12 x - 29 - f (x) > 0$

Step 2.1.3

Convert the inequality to an equation.

$- x^{2} + 12 x - 29 - f (x) = 0$

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 $a = - 1$ , $b = 12$ , and $c = - 29 - f (x)$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (- 1 \cdot (- 29 - f (x)))}}{2 \cdot - 1}$

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 $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 1 \cdot (- 29 - f (x))}}{2 \cdot - 1}$

Step 2.1.6.1.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{144 + 4 \cdot (- 29 - f (x))}}{2 \cdot - 1}$

Step 2.1.6.1.3

Apply the distributive property.

$x = \frac{- 12 \pm \sqrt{144 + 4 \cdot - 29 + 4 (- f (x))}}{2 \cdot - 1}$

Step 2.1.6.1.4

Multiply $4$ by $- 29$ .

$x = \frac{- 12 \pm \sqrt{144 - 116 + 4 (- f (x))}}{2 \cdot - 1}$

Step 2.1.6.1.5

Multiply $- 1$ by $4$ .

$x = \frac{- 12 \pm \sqrt{144 - 116 - 4 f x}}{2 \cdot - 1}$

Step 2.1.6.1.6

Subtract $116$ from $144$ .

$x = \frac{- 12 \pm \sqrt{28 - 4 f x}}{2 \cdot - 1}$

Step 2.1.6.1.7

Rewrite $28 - 4 f (x)$ as $2^{2} (7 - f (x))$ .

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Step 2.1.6.1.7.1

Factor $4$ out of $28$ .

$x = \frac{- 12 \pm \sqrt{4 (7) - 4 f x}}{2 \cdot - 1}$

Step 2.1.6.1.7.2

Factor $4$ out of $- 4 f (x)$ .

$x = \frac{- 12 \pm \sqrt{4 (7) + 4 (- f (x))}}{2 \cdot - 1}$

Step 2.1.6.1.7.3

Factor $4$ out of $4 (7) + 4 (- f (x))$ .

$x = \frac{- 12 \pm \sqrt{4 (7 - f (x))}}{2 \cdot - 1}$

Step 2.1.6.1.7.4

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 12 \pm \sqrt{2^{2} (7 - f (x))}}{2 \cdot - 1}$

Step 2.1.6.1.8

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{7 - f (x)}}{2 \cdot - 1}$

Step 2.1.6.2

Multiply $2$ by $- 1$ .

$x = \frac{- 12 \pm 2 \sqrt{7 - f (x)}}{- 2}$

Step 2.1.6.3

Simplify $\frac{- 12 \pm 2 \sqrt{7 - f (x)}}{- 2}$ .

$x = 6 \pm \sqrt{7 - f (x)}$

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 $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 1 \cdot (- 29 - f (x))}}{2 \cdot - 1}$

Step 2.1.7.1.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{144 + 4 \cdot (- 29 - f (x))}}{2 \cdot - 1}$

Step 2.1.7.1.3

Apply the distributive property.

$x = \frac{- 12 \pm \sqrt{144 + 4 \cdot - 29 + 4 (- f (x))}}{2 \cdot - 1}$

Step 2.1.7.1.4

Multiply $4$ by $- 29$ .

$x = \frac{- 12 \pm \sqrt{144 - 116 + 4 (- f (x))}}{2 \cdot - 1}$

Step 2.1.7.1.5

Multiply $- 1$ by $4$ .

$x = \frac{- 12 \pm \sqrt{144 - 116 - 4 f x}}{2 \cdot - 1}$

Step 2.1.7.1.6

Subtract $116$ from $144$ .

$x = \frac{- 12 \pm \sqrt{28 - 4 f x}}{2 \cdot - 1}$

Step 2.1.7.1.7

Rewrite $28 - 4 f (x)$ as $2^{2} (7 - f (x))$ .

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Step 2.1.7.1.7.1

Factor $4$ out of $28$ .

$x = \frac{- 12 \pm \sqrt{4 (7) - 4 f x}}{2 \cdot - 1}$

Step 2.1.7.1.7.2

Factor $4$ out of $- 4 f (x)$ .

$x = \frac{- 12 \pm \sqrt{4 (7) + 4 (- f (x))}}{2 \cdot - 1}$

Step 2.1.7.1.7.3

Factor $4$ out of $4 (7) + 4 (- f (x))$ .

$x = \frac{- 12 \pm \sqrt{4 (7 - f (x))}}{2 \cdot - 1}$

Step 2.1.7.1.7.4

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 12 \pm \sqrt{2^{2} (7 - f (x))}}{2 \cdot - 1}$

Step 2.1.7.1.8

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{7 - f (x)}}{2 \cdot - 1}$

Step 2.1.7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 12 \pm 2 \sqrt{7 - f (x)}}{- 2}$

Step 2.1.7.3

Simplify $\frac{- 12 \pm 2 \sqrt{7 - f (x)}}{- 2}$ .

$x = 6 \pm \sqrt{7 - f (x)}$

Step 2.1.7.4

Change the $\pm$ to $+$ .

$x = 6 + \sqrt{7 - f (x)}$

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 $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 1 \cdot (- 29 - f (x))}}{2 \cdot - 1}$

Step 2.1.8.1.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{144 + 4 \cdot (- 29 - f (x))}}{2 \cdot - 1}$

Step 2.1.8.1.3

Apply the distributive property.

$x = \frac{- 12 \pm \sqrt{144 + 4 \cdot - 29 + 4 (- f (x))}}{2 \cdot - 1}$

Step 2.1.8.1.4

Multiply $4$ by $- 29$ .

$x = \frac{- 12 \pm \sqrt{144 - 116 + 4 (- f (x))}}{2 \cdot - 1}$

Step 2.1.8.1.5

Multiply $- 1$ by $4$ .

$x = \frac{- 12 \pm \sqrt{144 - 116 - 4 f x}}{2 \cdot - 1}$

Step 2.1.8.1.6

Subtract $116$ from $144$ .

$x = \frac{- 12 \pm \sqrt{28 - 4 f x}}{2 \cdot - 1}$

Step 2.1.8.1.7

Rewrite $28 - 4 f (x)$ as $2^{2} (7 - f (x))$ .

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Step 2.1.8.1.7.1

Factor $4$ out of $28$ .

$x = \frac{- 12 \pm \sqrt{4 (7) - 4 f x}}{2 \cdot - 1}$

Step 2.1.8.1.7.2

Factor $4$ out of $- 4 f (x)$ .

$x = \frac{- 12 \pm \sqrt{4 (7) + 4 (- f (x))}}{2 \cdot - 1}$

Step 2.1.8.1.7.3

Factor $4$ out of $4 (7) + 4 (- f (x))$ .

$x = \frac{- 12 \pm \sqrt{4 (7 - f (x))}}{2 \cdot - 1}$

Step 2.1.8.1.7.4

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 12 \pm \sqrt{2^{2} (7 - f (x))}}{2 \cdot - 1}$

Step 2.1.8.1.8

Pull terms out from under the radical.

$x = \frac{- 12 \pm 2 \sqrt{7 - f (x)}}{2 \cdot - 1}$

Step 2.1.8.2

Multiply $2$ by $- 1$ .

$x = \frac{- 12 \pm 2 \sqrt{7 - f (x)}}{- 2}$

Step 2.1.8.3

Simplify $\frac{- 12 \pm 2 \sqrt{7 - f (x)}}{- 2}$ .

$x = 6 \pm \sqrt{7 - f (x)}$

Step 2.1.8.4

Change the $\pm$ to $-$ .

$x = 6 - \sqrt{7 - f (x)}$

Step 2.1.9

Consolidate the solutions.

$x = 6 + \sqrt{7 - f (x)}, 6 - \sqrt{7 - f (x)}$

Step 2.1.10

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

$- x^{2} + 12 x = f (x) + 29$

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 $- x^{2} + 12 x - 29 - f (x)$ .

$0 < - x^{2} + 12 x - 29 - f (x)$

Step 4