Precalculus Examples

$y + \frac{2}{5} x = 4$

Step 1

Write $y = 4 - \frac{2 x}{5}$ as a function.

$f (x) = 4 - \frac{2 x}{5}$

Step 2

Solve for $y$ .

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Step 2.1

Combine $\frac{2}{5}$ and $x$ .

$y + \frac{2 x}{5} = 4$

Step 2.2

Subtract $\frac{2 x}{5}$ from both sides of the equation.

$y = 4 - \frac{2 x}{5}$

Step 3

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 4

Find the components of the definition.

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Step 4.1

Evaluate the function at $x = x + h$ .

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Step 4.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = 4 - \frac{2 (x + h)}{5}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (x + h) = 4 \cdot \frac{5}{5} - \frac{2 (x + h)}{5}$

Step 4.1.2.2

Simplify terms.

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Step 4.1.2.2.1

Combine $4$ and $\frac{5}{5}$ .

$f (x + h) = \frac{4 \cdot 5}{5} - \frac{2 (x + h)}{5}$

Step 4.1.2.2.2

Combine the numerators over the common denominator.

$f (x + h) = \frac{4 \cdot 5 - 2 (x + h)}{5}$

Step 4.1.2.3

Simplify the numerator.

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Step 4.1.2.3.1

Factor $2$ out of $4 \cdot 5 - 2 (x + h)$ .

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Step 4.1.2.3.1.1

Factor $2$ out of $4 \cdot 5$ .

$f (x + h) = \frac{2 (2 \cdot 5) - 2 (x + h)}{5}$

Step 4.1.2.3.1.2

Factor $2$ out of $- 2 (x + h)$ .

$f (x + h) = \frac{2 (2 \cdot 5) + 2 (- (x + h))}{5}$

Step 4.1.2.3.1.3

Factor $2$ out of $2 (2 \cdot 5) + 2 (- (x + h))$ .

$f (x + h) = \frac{2 (2 \cdot 5 - (x + h))}{5}$

Step 4.1.2.3.2

Multiply $2$ by $5$ .

$f (x + h) = \frac{2 (10 - (x + h))}{5}$

Step 4.1.2.3.3

Apply the distributive property.

$f (x + h) = \frac{2 (10 - x - h)}{5}$

Step 4.1.2.4

The final answer is $\frac{2 (10 - x - h)}{5}$ .

$\frac{2 (10 - x - h)}{5}$

Step 4.2

Find the components of the definition.

$f (x + h) = \frac{2 (10 - x - h)}{5}$

$f (x) = 4 - \frac{2 x}{5}$

$f (x + h) = \frac{2 (10 - x - h)}{5}$

$f (x) = 4 - \frac{2 x}{5}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{2 (10 - x - h)}{5} - (4 - \frac{2 x}{5})}{h}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

$\frac{\frac{2 (10 - x - h)}{5} - 1 \cdot 4 - - \frac{2 x}{5}}{h}$

Step 6.1.2

Multiply $- 1$ by $4$ .

$\frac{\frac{2 (10 - x - h)}{5} - 4 - - \frac{2 x}{5}}{h}$

Step 6.1.3

Multiply $- - \frac{2 x}{5}$ .

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Step 6.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{2 (10 - x - h)}{5} - 4 + 1 \frac{2 x}{5}}{h}$

Step 6.1.3.2

Multiply $\frac{2 x}{5}$ by $1$ .

$\frac{\frac{2 (10 - x - h)}{5} - 4 + \frac{2 x}{5}}{h}$

Step 6.1.4

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\frac{2 (10 - x - h)}{5} - 4 \cdot \frac{5}{5} + \frac{2 x}{5}}{h}$

Step 6.1.5

Combine $- 4$ and $\frac{5}{5}$ .

$\frac{\frac{2 (10 - x - h)}{5} + \frac{- 4 \cdot 5}{5} + \frac{2 x}{5}}{h}$

Step 6.1.6

Combine the numerators over the common denominator.

$\frac{\frac{2 (10 - x - h) - 4 \cdot 5}{5} + \frac{2 x}{5}}{h}$

Step 6.1.7

Combine the numerators over the common denominator.

$\frac{\frac{2 (10 - x - h) - 4 \cdot 5 + 2 x}{5}}{h}$

Step 6.1.8

Simplify each term.

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Step 6.1.8.1

Apply the distributive property.

$\frac{\frac{2 \cdot 10 + 2 (- x) + 2 (- h) - 4 \cdot 5 + 2 x}{5}}{h}$

Step 6.1.8.2

Simplify.

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Step 6.1.8.2.1

Multiply $2$ by $10$ .

$\frac{\frac{20 + 2 (- x) + 2 (- h) - 4 \cdot 5 + 2 x}{5}}{h}$

Step 6.1.8.2.2

Multiply $- 1$ by $2$ .

$\frac{\frac{20 - 2 x + 2 (- h) - 4 \cdot 5 + 2 x}{5}}{h}$

Step 6.1.8.2.3

Multiply $- 1$ by $2$ .

$\frac{\frac{20 - 2 x - 2 h - 4 \cdot 5 + 2 x}{5}}{h}$

Step 6.1.8.3

Multiply $- 4$ by $5$ .

$\frac{\frac{20 - 2 x - 2 h - 20 + 2 x}{5}}{h}$

Step 6.1.9

Combine the opposite terms in $20 - 2 x - 2 h - 20 + 2 x$ .

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Step 6.1.9.1

Subtract $20$ from $20$ .

$\frac{\frac{- 2 x - 2 h + 0 + 2 x}{5}}{h}$

Step 6.1.9.2

Add $- 2 x - 2 h$ and $0$ .

$\frac{\frac{- 2 x - 2 h + 2 x}{5}}{h}$

Step 6.1.9.3

Add $- 2 x$ and $2 x$ .

$\frac{\frac{- 2 h + 0}{5}}{h}$

Step 6.1.9.4

Add $- 2 h$ and $0$ .

$\frac{\frac{- 2 h}{5}}{h}$

Step 6.1.10

Move the negative in front of the fraction.

$\frac{- \frac{2 h}{5}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{2 h}{5} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Move the leading negative in $- \frac{2 h}{5}$ into the numerator.

$\frac{- 2 h}{5} \cdot \frac{1}{h}$

Step 6.3.2

Factor $h$ out of $- 2 h$ .

$\frac{h \cdot - 2}{5} \cdot \frac{1}{h}$

Step 6.3.3

Cancel the common factor.

$\frac{h \cdot - 2}{5} \cdot \frac{1}{h}$

Step 6.3.4

Rewrite the expression.

$\frac{- 2}{5}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{2}{5}$

Step 7