Algebra Examples

$f (x) < 3 {(x - 2)}^{2} - 5$

Step 1

Solve for $y$ .

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Step 1.1

Simplify $3 {(x - 2)}^{2} - 5$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$f (x) < 3 ((x - 2) (x - 2)) - 5$

Step 1.1.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.1.1.2.1

Apply the distributive property.

$f (x) < 3 (x (x - 2) - 2 (x - 2)) - 5$

Step 1.1.1.2.2

Apply the distributive property.

$f (x) < 3 (x \cdot x + x \cdot - 2 - 2 (x - 2)) - 5$

Step 1.1.1.2.3

Apply the distributive property.

$f (x) < 3 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) - 5$

Step 1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.3.1

Simplify each term.

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Step 1.1.1.3.1.1

Multiply $x$ by $x$ .

$f (x) < 3 (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) - 5$

Step 1.1.1.3.1.2

Move $- 2$ to the left of $x$ .

$f (x) < 3 (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) - 5$

Step 1.1.1.3.1.3

Multiply $- 2$ by $- 2$ .

$f (x) < 3 (x^{2} - 2 x - 2 x + 4) - 5$

Step 1.1.1.3.2

Subtract $2 x$ from $- 2 x$ .

$f (x) < 3 (x^{2} - 4 x + 4) - 5$

Step 1.1.1.4

Apply the distributive property.

$f (x) < 3 x^{2} + 3 (- 4 x) + 3 \cdot 4 - 5$

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

Multiply $- 4$ by $3$ .

$f (x) < 3 x^{2} - 12 x + 3 \cdot 4 - 5$

Step 1.1.1.5.2

Multiply $3$ by $4$ .

$f (x) < 3 x^{2} - 12 x + 12 - 5$

Step 1.1.2

Subtract $5$ from $12$ .

$f (x) < 3 x^{2} - 12 x + 7$

Step 1.2

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

$0 < 3 x^{2} - 12 x + 7 - f (x)$

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.

$3 x^{2} - 12 x + 7 - f (x) > 0$

Step 2.1.3

Convert the inequality to an equation.

$3 x^{2} - 12 x + 7 - f (x) = 0$

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 $a = 3$ , $b = - 12$ , and $c = 7 - f (x)$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (3 \cdot (7 - f (x)))}}{2 \cdot 3}$

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 $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot (7 - f (x))}}{2 \cdot 3}$

Step 2.1.6.1.2

Multiply $- 4$ by $3$ .

$x = \frac{12 \pm \sqrt{144 - 12 \cdot (7 - f (x))}}{2 \cdot 3}$

Step 2.1.6.1.3

Apply the distributive property.

$x = \frac{12 \pm \sqrt{144 - 12 \cdot 7 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.6.1.4

Multiply $- 12$ by $7$ .

$x = \frac{12 \pm \sqrt{144 - 84 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.6.1.5

Multiply $- 1$ by $- 12$ .

$x = \frac{12 \pm \sqrt{144 - 84 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.6.1.6

Subtract $84$ from $144$ .

$x = \frac{12 \pm \sqrt{60 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.6.2

Multiply $2$ by $3$ .

$x = \frac{12 \pm \sqrt{60 + 12 f (x)}}{6}$

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 $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot (7 - f (x))}}{2 \cdot 3}$

Step 2.1.7.1.2

Multiply $- 4$ by $3$ .

$x = \frac{12 \pm \sqrt{144 - 12 \cdot (7 - f (x))}}{2 \cdot 3}$

Step 2.1.7.1.3

Apply the distributive property.

$x = \frac{12 \pm \sqrt{144 - 12 \cdot 7 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.7.1.4

Multiply $- 12$ by $7$ .

$x = \frac{12 \pm \sqrt{144 - 84 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.7.1.5

Multiply $- 1$ by $- 12$ .

$x = \frac{12 \pm \sqrt{144 - 84 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.7.1.6

Subtract $84$ from $144$ .

$x = \frac{12 \pm \sqrt{60 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.7.2

Multiply $2$ by $3$ .

$x = \frac{12 \pm \sqrt{60 + 12 f (x)}}{6}$

Step 2.1.7.3

Change the $\pm$ to $+$ .

$x = \frac{12 + \sqrt{60 + 12 f (x)}}{6}$

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 $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 3 \cdot (7 - f (x))}}{2 \cdot 3}$

Step 2.1.8.1.2

Multiply $- 4$ by $3$ .

$x = \frac{12 \pm \sqrt{144 - 12 \cdot (7 - f (x))}}{2 \cdot 3}$

Step 2.1.8.1.3

Apply the distributive property.

$x = \frac{12 \pm \sqrt{144 - 12 \cdot 7 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.8.1.4

Multiply $- 12$ by $7$ .

$x = \frac{12 \pm \sqrt{144 - 84 - 12 (- f (x))}}{2 \cdot 3}$

Step 2.1.8.1.5

Multiply $- 1$ by $- 12$ .

$x = \frac{12 \pm \sqrt{144 - 84 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.8.1.6

Subtract $84$ from $144$ .

$x = \frac{12 \pm \sqrt{60 + 12 f (x)}}{2 \cdot 3}$

Step 2.1.8.2

Multiply $2$ by $3$ .

$x = \frac{12 \pm \sqrt{60 + 12 f (x)}}{6}$

Step 2.1.8.3

Change the $\pm$ to $-$ .

$x = \frac{12 - \sqrt{60 + 12 f (x)}}{6}$

Step 2.1.9

Consolidate the solutions.

$x = \frac{12 + \sqrt{60 + 12 f (x)}}{6}, \frac{12 - \sqrt{60 + 12 f (x)}}{6}$

Step 2.1.10

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

$3 x^{2} - 12 x = f (x) - 7$

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 $3 x^{2} - 12 x + 7 - f (x)$ .

$0 < 3 x^{2} - 12 x + 7 - f (x)$

Step 4