Algebra Examples

$y = \frac{\sqrt{5 - x}}{\log (x)}$

Step 1

Set the argument in $\log (x)$ greater than $0$ to find where the expression is defined.

$x > 0$

Step 2

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

$5 - x \geq 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $5$ from both sides of the inequality.

$- x \geq - 5$

Step 3.2

Divide each term in $- x \geq - 5$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x \geq - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2

Divide $x$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 5$ by $- 1$ .

$x \leq 5$

Step 4

Set the denominator in $\frac{\sqrt{5 - x}}{\log (x)}$ equal to $0$ to find where the expression is undefined.

$\log (x) = 0$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite $\log (x) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{0} = x$

Step 5.2

Solve for $x$ .

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Step 5.2.1

Rewrite the equation as $x = 10^{0}$ .

$x = 10^{0}$

Step 5.2.2

Anything raised to $0$ is $1$ .

$x = 1$

Step 6

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

Interval Notation:

$(0, 1) \cup (1, 5]$

Set-Builder Notation:

${x | 0 < x \leq 5, x \neq 1}$

Step 7