Solve y=|4- 12/(x-4)

Question

Function

Evaluate the derivative

Find the domain

Find the x -intercept/zero

y^{'} = {\frac{12}{( x - 4 ) ^{2}}, x \in (- \infty, 4) \cup (7, + \infty) - \frac{12}{( x - 4 ) ^{2}}, 4 < x < 7

Evaluate

y = 4 - \frac{12}{x - 4}

Simplify

More Steps

Evaluate

4 - \frac{12}{x - 4}

Subtract the terms

More Steps

Simplify

4 - \frac{12}{x - 4}

Reduce fractions to a common denominator

\frac{4 ( x - 4 )}{x - 4} - \frac{12}{x - 4}

Write all numerators above the common denominator

\frac{4 ( x - 4 ) - 12}{x - 4}

Multiply the terms

\frac{4 x - 16 - 12}{x - 4}

Subtract the numbers

\frac{4 x - 28}{x - 4}

\frac{4 x - 28}{x - 4}

Rewrite the expression

4 \times \frac{x - 7}{x - 4}

Rewrite the expression

4 \frac{x - 7}{x - 4}

y = 4 \frac{x - 7}{x - 4}

Take the derivative of both sides

y^{'} = \frac{d}{d x} (4 \frac{x - 7}{x - 4})

Solution

y^{'} = {\frac{12}{( x - 4 ) ^{2}}, x \in (- \infty, 4) \cup (7, + \infty) - \frac{12}{( x - 4 ) ^{2}}, 4 < x < 7

Show Solution

Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

Evaluate

y = 4 - \frac{12}{x - 4}

Simplify the expression

y = 4 \frac{x - 7}{x - 4}

To test if the graph of y = 4 \frac{x - 7}{x - 4} is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 4 \frac{- x - 7}{- x - 4}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- y = 4 \frac{x + 7}{x + 4}

Change the signs both sides

y = - 4 \frac{x + 7}{x + 4}

Solution

Not symmetry with respect to the origin

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{28 - 4 y}{4 - y} x = \frac{28 + 4 y}{4 + y}

Evaluate

y = 4 - \frac{12}{x - 4}

Simplify

More Steps

Evaluate

4 - \frac{12}{x - 4}

Subtract the terms

More Steps

Simplify

4 - \frac{12}{x - 4}

Reduce fractions to a common denominator

\frac{4 ( x - 4 )}{x - 4} - \frac{12}{x - 4}

Write all numerators above the common denominator

\frac{4 ( x - 4 ) - 12}{x - 4}

Multiply the terms

\frac{4 x - 16 - 12}{x - 4}

Subtract the numbers

\frac{4 x - 28}{x - 4}

\frac{4 x - 28}{x - 4}

Rewrite the expression

4 \times \frac{x - 7}{x - 4}

Rewrite the expression

4 \frac{x - 7}{x - 4}

y = 4 \frac{x - 7}{x - 4}

Swap the sides of the equation

4 \frac{x - 7}{x - 4} = y

Divide both sides

\frac{4 \frac{x - 7}{x - 4}}{4} = \frac{y}{4}

Divide the numbers

\frac{x - 7}{x - 4} = \frac{y}{4}

Separate the equation into 2 possible cases

\frac{x - 7}{x - 4} = \frac{y}{4} \frac{x - 7}{x - 4} = - \frac{y}{4}

Solve the equation for x

More Steps

Evaluate

\frac{x - 7}{x - 4} = \frac{y}{4}

Cross multiply

(x - 7) \times 4 = (x - 4) y

Simplify the equation

4 (x - 7) = (x - 4) y

Simplify the equation

4 (x - 7) = y (x - 4)

Calculate

More Steps

Evaluate

4 (x - 7)

Apply the distributive property

4 x - 4 \times 7

Multiply the numbers

4 x - 28

4 x - 28 = y (x - 4)

Calculate

More Steps

Evaluate

y (x - 4)

Apply the distributive property

y x - y \times 4

Use the commutative property to reorder the terms

y x - 4 y

4 x - 28 = y x - 4 y

Move the expression to the left side

4 x - 28 - (y x - 4 y) = 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

4 x - 28 - y x + 4 y = 0

Collect like terms by calculating the sum or difference of their coefficients

(4 - y) x - 28 + 4 y = 0

Move the constant to the right side

(4 - y) x = 0 + 28 - 4 y

Removing 0 doesn't change the value,so remove it from the expression

(4 - y) x = 28 - 4 y

Divide both sides

\frac{( 4 - y ) x}{4 - y} = \frac{28 - 4 y}{4 - y}

Divide the numbers

x = \frac{28 - 4 y}{4 - y}

x = \frac{28 - 4 y}{4 - y} \frac{x - 7}{x - 4} = - \frac{y}{4}

Solution

More Steps

Evaluate

\frac{x - 7}{x - 4} = - \frac{y}{4}

Cross multiply

(x - 7) \times 4 = (x - 4) (- y)

Simplify the equation

4 (x - 7) = (x - 4) (- y)

Simplify the equation

4 (x - 7) = - y (x - 4)

Calculate

More Steps

Evaluate

4 (x - 7)

Apply the distributive property

4 x - 4 \times 7

Multiply the numbers

4 x - 28

4 x - 28 = - y (x - 4)

Calculate

More Steps

Evaluate

- y (x - 4)

Apply the distributive property

- y x - (- y \times 4)

Use the commutative property to reorder the terms

- y x - (- 4 y)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

- y x + 4 y

4 x - 28 = - y x + 4 y

Move the expression to the left side

4 x - 28 - (- y x + 4 y) = 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

4 x - 28 + y x - 4 y = 0

Collect like terms by calculating the sum or difference of their coefficients

(4 + y) x - 28 - 4 y = 0

Move the constant to the right side

(4 + y) x = 0 + 28 + 4 y

Removing 0 doesn't change the value,so remove it from the expression

(4 + y) x = 28 + 4 y

Divide both sides

\frac{( 4 + y ) x}{4 + y} = \frac{28 + 4 y}{4 + y}

Divide the numbers

x = \frac{28 + 4 y}{4 + y}

x = \frac{28 - 4 y}{4 - y} x = \frac{28 + 4 y}{4 + y}

Show Solution

Function

Testing for symmetry

Solve the equation

Graph