Precalculus Examples

$x > - \frac{3}{x - 4}$

Step 1

Add $\frac{3}{x - 4}$ to both sides of the inequality.

$x + \frac{3}{x - 4} > 0$

Step 2

Simplify $x + \frac{3}{x - 4}$ .

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Step 2.1

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

$\frac{x (x - 4)}{x - 4} + \frac{3}{x - 4} > 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{x (x - 4) + 3}{x - 4} > 0$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4 + 3}{x - 4} > 0$

Step 2.3.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 4 + 3}{x - 4} > 0$

Step 2.3.3

Move $- 4$ to the left of $x$ .

$\frac{x^{2} - 4 \cdot x + 3}{x - 4} > 0$

Step 2.3.4

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 2.3.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 2.3.4.2

Write the factored form using these integers.

$\frac{(x - 3) (x - 1)}{x - 4} > 0$

Step 3

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

$x - 3 = 0$

$x - 1 = 0$

$x - 4 = 0$

Step 4

Add $3$ to both sides of the equation.

$x = 3$

Step 5

Add $1$ to both sides of the equation.

$x = 1$

Step 6

Add $4$ to both sides of the equation.

$x = 4$

Step 7

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

$x = 3$

$x = 1$

$x = 4$

Step 8

Consolidate the solutions.

$x = 3, 1, 4$

Step 9

Find the domain of $\frac{(x - 3) (x - 1)}{x - 4}$ .

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Step 9.1

Set the denominator in $\frac{(x - 3) (x - 1)}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 9.2

Add $4$ to both sides of the equation.

$x = 4$

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.

$x < 1$

$1 < x < 3$

$3 < x < 4$

$x > 4$

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

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Step 11.1.1

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

$x = 0$

Step 11.1.2

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

$0 > - \frac{3}{(0) - 4}$

Step 11.1.3

The left side $0$ is not greater than the right side $0.75$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $1 < x < 3$ to see if it makes the inequality true.

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Step 11.2.1

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

$x = 2$

Step 11.2.2

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

$2 > - \frac{3}{(2) - 4}$

Step 11.2.3

The left side $2$ is greater than the right side $1.5$ , which means that the given statement is always true.

True

Step 11.3

Test a value on the interval $3 < x < 4$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $3 < x < 4$ and see if this value makes the original inequality true.

$x = 3.5$

Step 11.3.2

Replace $x$ with $3.5$ in the original inequality.

$3.5 > - \frac{3}{(3.5) - 4}$

Step 11.3.3

The left side $3.5$ is not greater than the right side $6$ , which means that the given statement is false.

False

Step 11.4

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

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Step 11.4.1

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

$x = 6$

Step 11.4.2

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

$6 > - \frac{3}{(6) - 4}$

Step 11.4.3

The left side $6$ is greater than the right side $- 1.5$ , which means that the given statement is always true.

True

Step 11.5

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

$x < 1$ False

$1 < x < 3$ True

$3 < x < 4$ False

$x > 4$ True

$x < 1$ False

$1 < x < 3$ True

$3 < x < 4$ False

$x > 4$ True

Step 12

The solution consists of all of the true intervals.

$1 < x < 3$ or $x > 4$

Step 13

Convert the inequality to interval notation.

$(1, 3) \cup (4, \infty)$

Step 14