Precalculus Examples

$\frac{2 x^{2}}{3 x^{2} - 4 x}$

Step 1

Set the denominator in $\frac{2 x^{2}}{3 x^{2} - 4 x}$ equal to $0$ to find where the expression is undefined.

$3 x^{2} - 4 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x$ out of $3 x^{2} - 4 x$ .

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Step 2.1.1

Factor $x$ out of $3 x^{2}$ .

$x (3 x) - 4 x = 0$

Step 2.1.2

Factor $x$ out of $- 4 x$ .

$x (3 x) + x \cdot - 4 = 0$

Step 2.1.3

Factor $x$ out of $x (3 x) + x \cdot - 4$ .

$x (3 x - 4) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$3 x - 4 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $3 x - 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 2.4.2

Solve $3 x - 4 = 0$ for $x$ .

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Step 2.4.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 2.4.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 2.4.2.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 2.5

The final solution is all the values that make $x (3 x - 4) = 0$ true.

$x = 0, \frac{4}{3}$

Step 3

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

Interval Notation:

$(- \infty, 0) \cup (0, \frac{4}{3}) \cup (\frac{4}{3}, \infty)$

Set-Builder Notation:

${x | x \neq 0, \frac{4}{3}}$

Step 4