Precalculus Examples

$f (x) = \sqrt{\frac{1 - x}{2 x - 3}}$

Step 1

Set the radicand in $\sqrt{\frac{1 - x}{2 x - 3}}$ greater than or equal to $0$ to find where the expression is defined.

$\frac{1 - x}{2 x - 3} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$1 - x = 0$

$2 x - 3 = 0$

Step 2.2

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2

Divide $x$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 2.4

Add $3$ to both sides of the equation.

$2 x = 3$

Step 2.5

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 2.5.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

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.

$\frac{2 x}{2} = \frac{3}{2}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 2.6

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

$x = 1$

$x = \frac{3}{2}$

Step 2.7

Consolidate the solutions.

$x = 1, \frac{3}{2}$

Step 2.8

Find the domain of $\frac{1 - x}{2 x - 3}$ .

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Step 2.8.1

Set the denominator in $\frac{1 - x}{2 x - 3}$ equal to $0$ to find where the expression is undefined.

$2 x - 3 = 0$

Step 2.8.2

Solve for $x$ .

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Step 2.8.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 2.8.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 2.8.2.2.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

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.

$\frac{2 x}{2} = \frac{3}{2}$

Step 2.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 2.8.3

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

$(- \infty, \frac{3}{2}) \cup (\frac{3}{2}, \infty)$

Step 2.9

Use each root to create test intervals.

$x < 1$

$1 < x < \frac{3}{2}$

$x > \frac{3}{2}$

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 $x < 1$ to see if it makes the inequality true.

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Step 2.10.1.1

Choose a value on the interval $x < 1$ 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.

$\frac{1 - (0)}{2 (0) - 3} \geq 0$

Step 2.10.1.3

The left side $- 0.\overline{3}$ 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 $1 < x < \frac{3}{2}$ to see if it makes the inequality true.

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Step 2.10.2.1

Choose a value on the interval $1 < x < \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 1.25$

Step 2.10.2.2

Replace $x$ with $1.25$ in the original inequality.

$\frac{1 - (1.25)}{2 (1.25) - 3} \geq 0$

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 $x > \frac{3}{2}$ to see if it makes the inequality true.

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Step 2.10.3.1

Choose a value on the interval $x > \frac{3}{2}$ 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.

$\frac{1 - (4)}{2 (4) - 3} \geq 0$

Step 2.10.3.3

The left side $- 0.6$ 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.

$x < 1$ False

$1 < x < \frac{3}{2}$ True

$x > \frac{3}{2}$ False

$x < 1$ False

$1 < x < \frac{3}{2}$ True

$x > \frac{3}{2}$ False

Step 2.11

The solution consists of all of the true intervals.

$1 \leq x < \frac{3}{2}$

Step 3

Set the denominator in $\frac{1 - x}{2 x - 3}$ equal to $0$ to find where the expression is undefined.

$2 x - 3 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 4.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 4.2.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

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.

$\frac{2 x}{2} = \frac{3}{2}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 5

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

Interval Notation:

$[1, \frac{3}{2})$

Set-Builder Notation:

${x | 1 \leq x < \frac{3}{2}}$

Step 6