Precalculus Examples

$\frac{4 + x}{4 - x} \geq 1$

Step 1

Subtract $1$ from both sides of the inequality.

$\frac{4 + x}{4 - x} - 1 \geq 0$

Step 2

Simplify $\frac{4 + x}{4 - x} - 1$ .

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Step 2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4 - x}{4 - x}$ .

$\frac{4 + x}{4 - x} - 1 \cdot \frac{4 - x}{4 - x} \geq 0$

Step 2.2

Combine $- 1$ and $\frac{4 - x}{4 - x}$ .

$\frac{4 + x}{4 - x} + \frac{- (4 - x)}{4 - x} \geq 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{4 + x - (4 - x)}{4 - x} \geq 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{4 + x - 1 \cdot 4 - - x}{4 - x} \geq 0$

Step 2.4.2

Multiply $- 1$ by $4$ .

$\frac{4 + x - 4 - - x}{4 - x} \geq 0$

Step 2.4.3

Multiply $- - x$ .

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Step 2.4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{4 + x - 4 + 1 x}{4 - x} \geq 0$

Step 2.4.3.2

Multiply $x$ by $1$ .

$\frac{4 + x - 4 + x}{4 - x} \geq 0$

Step 2.4.4

Subtract $4$ from $4$ .

$\frac{x + 0 + x}{4 - x} \geq 0$

Step 2.4.5

Add $x$ and $0$ .

$\frac{x + x}{4 - x} \geq 0$

Step 2.4.6

Add $x$ and $x$ .

$\frac{2 x}{4 - x} \geq 0$

Step 3

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

$2 x = 0$

$4 - x = 0$

Step 4

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$x = 0$

Step 5

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 6

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 6.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 7

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

$x = 0$

$x = 4$

Step 8

Consolidate the solutions.

$x = 0, 4$

Step 9

Find the domain of $\frac{2 x}{4 - x}$ .

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Step 9.1

Set the denominator in $\frac{2 x}{4 - x}$ equal to $0$ to find where the expression is undefined.

$4 - x = 0$

Step 9.2

Solve for $x$ .

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Step 9.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 9.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 9.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 9.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 9.2.2.3

Simplify the right side.

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Step 9.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 9.3

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

$(- \infty, 4) \cup (4, \infty)$

Step 10

Use each root to create test intervals.

$x < 0$

$0 < x < 4$

$x > 4$

Step 11

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 11.1

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

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Step 11.1.1

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

$x = - 2$

Step 11.1.2

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

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

Step 11.1.3

The left side $0.\overline{3}$ is less than the right side $1$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 11.2.2

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

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

Step 11.2.3

The left side $3$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 11.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 11.3.2

Replace $x$ with $6$ in the original inequality.

$\frac{4 + 6}{4 - (6)} \geq 1$

Step 11.3.3

The left side $- 5$ is less than the right side $1$ , which means that the given statement is false.

False

Step 11.4

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

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

Step 12

The solution consists of all of the true intervals.

$0 \leq x < 4$

Step 13

Convert the inequality to interval notation.

$[0, 4)$

Step 14