Pre-Algebra Examples

$f (x) < \frac{525}{x} + 3$

Step 1

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

$0 < \frac{525}{x} + 3 - f (x)$

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.

$\frac{525}{x} + 3 - f (x) > 0$

Step 2.1.3

Move all terms not containing $x$ to the right side of the inequality.

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Step 2.1.3.1

Subtract $3$ from both sides of the inequality.

$\frac{525}{x} - f (x) > - 3$

Step 2.1.3.2

Add $f (x)$ to both sides of the inequality.

$\frac{525}{x} > - 3 + f (x)$

Step 2.1.4

Multiply both sides by $x$ .

$\frac{525}{x} x = (- 3 + f (x)) x$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Simplify the left side.

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Step 2.1.5.1.1

Cancel the common factor of $x$ .

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Step 2.1.5.1.1.1

Cancel the common factor.

$\frac{525}{x} x = (- 3 + f (x)) x$

Step 2.1.5.1.1.2

Rewrite the expression.

$525 = (- 3 + f (x)) x$

Step 2.1.5.2

Simplify the right side.

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Step 2.1.5.2.1

Simplify $(- 3 + f (x)) x$ .

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Step 2.1.5.2.1.1

Apply the distributive property.

$525 = - 3 x + f (x) x$

Step 2.1.5.2.1.2

Reorder factors in $- 3 x + f (x) x$ .

$525 = - 3 x + x f (x)$

Step 2.1.6

Solve for $x$ .

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Step 2.1.6.1

Rewrite the equation as $- 3 x + x f (x) = 525$ .

$- 3 x + x f (x) = 525$

Step 2.1.6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.6.2.1

Move $x$ .

$- 3 x + x \cdot x f = 525$

Step 2.1.6.2.2

Multiply $x$ by $x$ .

$- 3 x + x^{2} f = 525$

Step 2.1.6.3

Subtract $525$ from both sides of the equation.

$- 3 x + x^{2} f - 525 = 0$

Step 2.1.6.4

Use the quadratic formula to find the solutions.

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

Step 2.1.6.5

Substitute the values $a = f$ , $b = - 3$ , and $c = - 525$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (f \cdot - 525)}}{2 f}$

Step 2.1.6.6

Simplify the numerator.

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Step 2.1.6.6.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot f \cdot - 525}}{2 f}$

Step 2.1.6.6.2

Multiply $- 525$ by $- 4$ .

$x = \frac{3 \pm \sqrt{9 + 2100 f}}{2 f}$

Step 2.1.6.6.3

Factor $3$ out of $9 + 2100 f$ .

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Step 2.1.6.6.3.1

Factor $3$ out of $9$ .

$x = \frac{3 \pm \sqrt{3 (3) + 2100 f}}{2 f}$

Step 2.1.6.6.3.2

Factor $3$ out of $2100 f$ .

$x = \frac{3 \pm \sqrt{3 (3) + 3 (700 f)}}{2 f}$

Step 2.1.6.6.3.3

Factor $3$ out of $3 (3) + 3 (700 f)$ .

$x = \frac{3 \pm \sqrt{3 (3 + 700 f)}}{2 f}$

Step 2.1.6.7

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{3 (3 + 700 f)}}{2 f}$

Step 2.1.6.8

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

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Step 2.1.6.8.1

Simplify the numerator.

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Step 2.1.6.8.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot f \cdot - 525}}{2 f}$

Step 2.1.6.8.1.2

Multiply $- 525$ by $- 4$ .

$x = \frac{3 \pm \sqrt{9 + 2100 f}}{2 f}$

Step 2.1.6.8.1.3

Factor $3$ out of $9 + 2100 f$ .

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Step 2.1.6.8.1.3.1

Factor $3$ out of $9$ .

$x = \frac{3 \pm \sqrt{3 (3) + 2100 f}}{2 f}$

Step 2.1.6.8.1.3.2

Factor $3$ out of $2100 f$ .

$x = \frac{3 \pm \sqrt{3 (3) + 3 (700 f)}}{2 f}$

Step 2.1.6.8.1.3.3

Factor $3$ out of $3 (3) + 3 (700 f)$ .

$x = \frac{3 \pm \sqrt{3 (3 + 700 f)}}{2 f}$

Step 2.1.6.8.2

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{3 (3 + 700 f)}}{2 f}$

Step 2.1.6.9

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{3 (3 + 700 f)}}{2 f}$

$x = \frac{3 - \sqrt{3 (3 + 700 f)}}{2 f}$

$x = \frac{3 + \sqrt{3 (3 + 700 f)}}{2 f}$

$x = \frac{3 - \sqrt{3 (3 + 700 f)}}{2 f}$

Step 2.1.7

Rewrite in slope-intercept form.

$= \frac{3 + \sqrt{3 (3 + 700)}}{2}$

$= f + \frac{3 - \sqrt{3 (3 + 700 f)}}{2 f}$

$= \frac{3 + \sqrt{3 (3 + 700)}}{2}$

$= f + \frac{3 - \sqrt{3 (3 + 700 f)}}{2 f}$

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

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $\frac{525}{x} + 3 - f (x)$ .

$0 < \frac{525}{x} + 3 - f (x)$

Step 4