Finite Math Examples

$y = \frac{\sqrt[4]{5 - 10 x}}{(x - 3) (x + 4)}$

Step 1

Set the radicand in $\sqrt[4]{5 - 10 x}$ greater than or equal to $0$ to find where the expression is defined.

$5 - 10 x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $5$ from both sides of the inequality.

$- 10 x \geq - 5$

Step 2.2

Divide each term in $- 10 x \geq - 5$ by $- 10$ and simplify.

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Step 2.2.1

Divide each term in $- 10 x \geq - 5$ by $- 10$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 10 x}{- 10} \leq \frac{- 5}{- 10}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 10$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 10 x}{- 10} \leq \frac{- 5}{- 10}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 5}{- 10}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $- 5$ and $- 10$ .

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Step 2.2.3.1.1

Factor $- 5$ out of $- 5$ .

$x \leq \frac{- 5 (1)}{- 10}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $- 5$ out of $- 10$ .

$x \leq \frac{- 5 \cdot 1}{- 5 \cdot 2}$

Step 2.2.3.1.2.2

Cancel the common factor.

$x \leq \frac{- 5 \cdot 1}{- 5 \cdot 2}$

Step 2.2.3.1.2.3

Rewrite the expression.

$x \leq \frac{1}{2}$

Step 3

Set the denominator in $\frac{\sqrt[4]{5 - 10 x}}{(x - 3) (x + 4)}$ equal to $0$ to find where the expression is undefined.

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

Step 4

Solve for $x$ .

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Step 4.1

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

$x - 3 = 0$

$x + 4 = 0$

Step 4.2

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

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Step 4.2.1

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

$x - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.3

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 4.3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4.4

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

$x = 3, - 4$

Step 5

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

Interval Notation:

$(- \infty, - 4) \cup (- 4, \frac{1}{2}]$

Set-Builder Notation:

${x | x \leq \frac{1}{2}, x \neq - 4}$

Step 6