Precalculus Examples

$f (x) = \sqrt{\frac{x}{2 x - 1} - 1}$

Step 1

Set the radicand in $\sqrt{\frac{x}{2 x - 1} - 1}$ greater than or equal to $0$ to find where the expression is defined.

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

Step 2

Solve for $x$ .

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Step 2.1

Simplify $\frac{x}{2 x - 1} - 1$ .

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Step 2.1.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{x}{2 x - 1} - 1 \cdot \frac{2 x - 1}{2 x - 1} \geq 0$

Step 2.1.2

Simplify terms.

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Step 2.1.2.1

Combine $- 1$ and $\frac{2 x - 1}{2 x - 1}$ .

$\frac{x}{2 x - 1} + \frac{- (2 x - 1)}{2 x - 1} \geq 0$

Step 2.1.2.2

Combine the numerators over the common denominator.

$\frac{x - (2 x - 1)}{2 x - 1} \geq 0$

Step 2.1.3

Simplify the numerator.

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Step 2.1.3.1

Apply the distributive property.

$\frac{x - (2 x) - - 1}{2 x - 1} \geq 0$

Step 2.1.3.2

Multiply $2$ by $- 1$ .

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

Step 2.1.3.3

Multiply $- 1$ by $- 1$ .

$\frac{x - 2 x + 1}{2 x - 1} \geq 0$

Step 2.1.3.4

Subtract $2 x$ from $x$ .

$\frac{- x + 1}{2 x - 1} \geq 0$

Step 2.1.4

Simplify with factoring out.

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Step 2.1.4.1

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 1}{2 x - 1} \geq 0$

Step 2.1.4.2

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (x) - 1 (- 1)}{2 x - 1} \geq 0$

Step 2.1.4.3

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$\frac{- (x - 1)}{2 x - 1} \geq 0$

Step 2.1.4.4

Simplify the expression.

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Step 2.1.4.4.1

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$\frac{- 1 (x - 1)}{2 x - 1} \geq 0$

Step 2.1.4.4.2

Move the negative in front of the fraction.

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

Step 2.2

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

$x - 1 = 0$

$2 x - 1 = 0$

Step 2.3

Add $1$ to both sides of the equation.

$x = 1$

Step 2.4

Add $1$ to both sides of the equation.

$2 x = 1$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.1.1

Cancel the common factor.

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

Step 2.5.2.1.2

Divide $x$ by $1$ .

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

Step 2.6

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 1$

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

Step 2.7

Consolidate the solutions.

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

Step 2.8

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

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Step 2.8.1

Set the denominator in $\frac{x}{2 x - 1}$ equal to $0$ to find where the expression is undefined.

$2 x - 1 = 0$

Step 2.8.2

Solve for $x$ .

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Step 2.8.2.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 2.8.2.2

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

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Step 2.8.2.2.1

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

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

Step 2.8.2.2.2

Simplify the left side.

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Step 2.8.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.8.2.2.2.1.1

Cancel the common factor.

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

Step 2.8.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.8.3

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

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

Step 2.9

Use each root to create test intervals.

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

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

$x > 1$

Step 2.10

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 2.10.1

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

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Step 2.10.1.1

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

$x = 0$

Step 2.10.1.2

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

$\frac{0}{2 (0) - 1} - 1 \geq 0$

Step 2.10.1.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.10.2

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

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Step 2.10.2.1

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

$x = 0.75$

Step 2.10.2.2

Replace $x$ with $0.75$ in the original inequality.

$\frac{0.75}{2 (0.75) - 1} - 1 \geq 0$

Step 2.10.2.3

The left side $0.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.10.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 2.10.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 2.10.3.2

Replace $x$ with $4$ in the original inequality.

$\frac{4}{2 (4) - 1} - 1 \geq 0$

Step 2.10.3.3

The left side $- 0.\overline{428571}$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.10.4

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

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

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

$x > 1$ False

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

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

$x > 1$ False

Step 2.11

The solution consists of all of the true intervals.

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

Step 3

Set the denominator in $\frac{x}{2 x - 1}$ equal to $0$ to find where the expression is undefined.

$2 x - 1 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

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

Step 5

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

Interval Notation:

$(\frac{1}{2}, 1]$

Set-Builder Notation:

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

Step 6