Algebra Examples

$y = \frac{2 x - 11}{3 - 5 x}$

Step 1

Set the denominator in $\frac{2 x - 11}{3 - 5 x}$ equal to $0$ to find where the expression is undefined.

$3 - 5 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $3$ from both sides of the equation.

$- 5 x = - 3$

Step 2.2

Divide each term in $- 5 x = - 3$ by $- 5$ and simplify.

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Step 2.2.1

Divide each term in $- 5 x = - 3$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 3}{- 5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 3}{- 5}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 5}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{3}{5}$

Step 3

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

Interval Notation:

$(- \infty, \frac{3}{5}) \cup (\frac{3}{5}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{3}{5}}$

Step 4

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

Interval Notation:

$(- \infty, - \frac{2}{5}) \cup (- \frac{2}{5}, \infty)$

Set-Builder Notation:

${y | y \neq - \frac{2}{5}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{3}{5}) \cup (\frac{3}{5}, \infty), {x | x \neq \frac{3}{5}}$

Range: $(- \infty, - \frac{2}{5}) \cup (- \frac{2}{5}, \infty), {y | y \neq - \frac{2}{5}}$

Step 6