Find the Domain and Range (6x+7)/(5x-6)

$\frac{6 x + 7}{5 x - 6}$

Step 1

Set the denominator in $\frac{6 x + 7}{5 x - 6}$ equal to $0$ to find where the expression is undefined.

$5 x - 6 = 0$

Step 2

Solve for $x$ .

Add $6$ to both sides of the equation.

$5 x = 6$

Divide each term in $5 x = 6$ by $5$ and simplify.

Divide each term in $5 x = 6$ by $5$ .

$\frac{5 x}{5} = \frac{6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{6}{5}$

Divide $x$ by $1$ .

$x = \frac{6}{5}$

Step 3

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

Interval Notation:

$(- \infty, \frac{6}{5}) \cup (\frac{6}{5}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{6}{5}}$

Step 4

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

Interval Notation:

$(- \infty, \frac{6}{5}) \cup (\frac{6}{5}, \infty)$

Set-Builder Notation:

${y | y \neq \frac{6}{5}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{6}{5}) \cup (\frac{6}{5}, \infty), {x | x \neq \frac{6}{5}}$

Range: $(- \infty, \frac{6}{5}) \cup (\frac{6}{5}, \infty), {y | y \neq \frac{6}{5}}$

Step 6