Finite Math Examples

$4 \log_{5} (\sqrt{6 x - 1}) - \log_{5} (\frac{5}{x}) + \log_{5} (5)$

Step 1

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

$x = 0$

Step 2

Set the argument in $\log_{5} (\sqrt{6 x - 1})$ less than or equal to $0$ to find where the expression is undefined.

$\sqrt{6 x - 1} \leq 0$

Step 3

Solve for $x$ .

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Step 3.1

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{6 x - 1}}^{2} \leq 0^{2}$

Step 3.2

Simplify each side of the inequality.

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Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x - 1}$ as ${(6 x - 1)}^{\frac{1}{2}}$ .

${({(6 x - 1)}^{\frac{1}{2}})}^{2} \leq 0^{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify ${({(6 x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.2.1.1

Multiply the exponents in ${({(6 x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(6 x - 1)}^{\frac{1}{2} \cdot 2} \leq 0^{2}$

Step 3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.2.1

Cancel the common factor.

${(6 x - 1)}^{\frac{1}{2} \cdot 2} \leq 0^{2}$

Step 3.2.2.1.1.2.2

Rewrite the expression.

${(6 x - 1)}^{1} \leq 0^{2}$

Step 3.2.2.1.2

Simplify.

$6 x - 1 \leq 0^{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raising $0$ to any positive power yields $0$ .

$6 x - 1 \leq 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

Add $1$ to both sides of the inequality.

$6 x \leq 1$

Step 3.3.2

Divide each term in $6 x \leq 1$ by $6$ and simplify.

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Step 3.3.2.1

Divide each term in $6 x \leq 1$ by $6$ .

$\frac{6 x}{6} \leq \frac{1}{6}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $6$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} \leq \frac{1}{6}$

Step 3.3.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{1}{6}$

Step 3.4

Find the domain of $\sqrt{6 x - 1}$ .

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Step 3.4.1

Set the radicand in $\sqrt{6 x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$6 x - 1 \geq 0$

Step 3.4.2

Solve for $x$ .

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Step 3.4.2.1

Add $1$ to both sides of the inequality.

$6 x \geq 1$

Step 3.4.2.2

Divide each term in $6 x \geq 1$ by $6$ and simplify.

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Step 3.4.2.2.1

Divide each term in $6 x \geq 1$ by $6$ .

$\frac{6 x}{6} \geq \frac{1}{6}$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $6$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} \geq \frac{1}{6}$

Step 3.4.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{1}{6}$

Step 3.4.3

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

$[\frac{1}{6}, \infty)$

Step 3.5

Use each root to create test intervals.

$x < \frac{1}{6}$

$x > \frac{1}{6}$

Step 3.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 3.6.1

Test a value on the interval $x < \frac{1}{6}$ to see if it makes the inequality true.

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Step 3.6.1.1

Choose a value on the interval $x < \frac{1}{6}$ and see if this value makes the original inequality true.

$x = 0$

Step 3.6.1.2

Replace $x$ with $0$ in the original inequality.

$\sqrt{6 (0) - 1} \leq 0$

Step 3.6.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 3.6.2

Test a value on the interval $x > \frac{1}{6}$ to see if it makes the inequality true.

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Step 3.6.2.1

Choose a value on the interval $x > \frac{1}{6}$ and see if this value makes the original inequality true.

$x = 3$

Step 3.6.2.2

Replace $x$ with $3$ in the original inequality.

$\sqrt{6 (3) - 1} \leq 0$

Step 3.6.2.3

The left side $4.12310562$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 3.6.3

Compare the intervals to determine which ones satisfy the original inequality.

$x < \frac{1}{6}$ False

$x > \frac{1}{6}$ False

$x < \frac{1}{6}$ False

$x > \frac{1}{6}$ False

Step 3.7

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 4

Set the argument in $\log_{5} (\frac{5}{x})$ less than or equal to $0$ to find where the expression is undefined.

$\frac{5}{x} \leq 0$

Step 5

Solve for $x$ .

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Step 5.1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x = 0$

Step 5.2

Find the domain of $\frac{5}{x}$ .

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Step 5.2.1

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

$x = 0$

Step 5.2.2

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

$(- \infty, 0) \cup (0, \infty)$

Step 5.3

The solution consists of all of the true intervals.

$x < 0$

Step 6

Set the radicand in $\sqrt{6 x - 1}$ less than $0$ to find where the expression is undefined.

$6 x - 1 < 0$

Step 7

Solve for $x$ .

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Step 7.1

Add $1$ to both sides of the inequality.

$6 x < 1$

Step 7.2

Divide each term in $6 x < 1$ by $6$ and simplify.

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Step 7.2.1

Divide each term in $6 x < 1$ by $6$ .

$\frac{6 x}{6} < \frac{1}{6}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $6$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} < \frac{1}{6}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{1}{6}$

Step 8

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x < \frac{1}{6}$

$(- \infty, \frac{1}{6})$

Step 9