Pre-Algebra Examples

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

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

Step 1

Subtract

f (x)

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

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

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

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

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 2.1.2

Rewrite so

x

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

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

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

Step 2.1.3

Move all terms not containing

x

$x$ to the right side of the inequality.

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Step 2.1.3.1

Subtract

3

$3$ from both sides of the inequality.

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

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

Step 2.1.3.2

Add

f (x)

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

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

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

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

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

Step 2.1.4

Multiply both sides by

x

$x$ .

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

$x$ .

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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