Algebra Examples

$y = 10 + \frac{10}{4 x + 12}$

Step 1

Set the denominator in $\frac{10}{4 x + 12}$ equal to $0$ to find where the expression is undefined.

$4 x + 12 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $12$ from both sides of the equation.

$4 x = - 12$

Step 2.2

Divide each term in $4 x = - 12$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 x = - 12$ by $4$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 12$ by $4$ .

$x = - 3$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 3) \cup (- 3, \infty)$

Set-Builder Notation:

${x | x \neq - 3}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, 10) \cup (10, \infty)$

Set-Builder Notation:

${y | y \neq 10}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 3) \cup (- 3, \infty), {x | x \neq - 3}$

Range: $(- \infty, 10) \cup (10, \infty), {y | y \neq 10}$

Step 6