Precalculus Examples

$\log_{4} (x^{- 1} - 2) = 1$

Step 1

Set the argument in $\log_{4} (x^{- 1} - 2)$ greater than $0$ to find where the expression is defined.

$x^{- 1} - 2 > 0$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x} - 2 > 0$

Step 2.2

Add $2$ to both sides of the inequality.

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

Step 2.3

Multiply both sides by $x$ .

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

Step 2.4

Simplify the left side.

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Step 2.4.1

Cancel the common factor of $x$ .

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Step 2.4.1.1

Cancel the common factor.

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

Step 2.4.1.2

Rewrite the expression.

$1 = 2 x$

Step 2.5

Solve for $x$ .

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Step 2.5.1

Rewrite the equation as $2 x = 1$ .

$2 x = 1$

Step 2.5.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 2.5.2.1

Divide each term in $2 x = 1$ by $2$ .

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

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

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

Step 2.5.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.6

Find the domain of $x^{- 1} - 2$ .

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Step 2.6.1

Set the base in $x^{- 1}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 2.6.2

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

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

Step 2.7

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{1}{2}$

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

Step 2.8

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 2.8.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 2.8.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 2.8.1.2

Replace $x$ with $- 2$ in the original inequality.

${(- 2)}^{- 1} - 2 > 0$

Step 2.8.1.3

The left side $- 2.5$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 2.8.2

Test a value on the interval $0 < x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 2.8.2.1

Choose a value on the interval $0 < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0.25$

Step 2.8.2.2

Replace $x$ with $0.25$ in the original inequality.

${(0.25)}^{- 1} - 2 > 0$

Step 2.8.2.3

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.8.3

Test a value on the interval $x > \frac{1}{2}$ to see if it makes the inequality true.

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Step 2.8.3.1

Choose a value on the interval $x > \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 3$

Step 2.8.3.2

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

${(3)}^{- 1} - 2 > 0$

Step 2.8.3.3

The left side $- 1.\overline{6}$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 2.8.4

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

$x < 0$ False

$0 < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

$x < 0$ False

$0 < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

Step 2.9

The solution consists of all of the true intervals.

$0 < x < \frac{1}{2}$

Step 3

Set the base in $x^{- 1}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4

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

Interval Notation:

$(0, \frac{1}{2})$

Set-Builder Notation:

${x | 0 < x < \frac{1}{2}}$

Step 5