Trigonometry Examples

$\frac{4 x - 8}{x - 7} \geq 0$

Step 1

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

$4 x - 8 = 0$

$x - 7 = 0$

Step 2

Add $8$ to both sides of the equation.

$4 x = 8$

Step 3

Divide each term in $4 x = 8$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 x = 8$ by $4$ .

$\frac{4 x}{4} = \frac{8}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{8}{4}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $8$ by $4$ .

$x = 2$

Step 4

Add $7$ to both sides of the equation.

$x = 7$

Step 5

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

$x = 2$

$x = 7$

Step 6

Consolidate the solutions.

$x = 2, 7$

Step 7

Find the domain of $\frac{4 x - 8}{x - 7}$ .

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Step 7.1

Set the denominator in $\frac{4 x - 8}{x - 7}$ equal to $0$ to find where the expression is undefined.

$x - 7 = 0$

Step 7.2

Add $7$ to both sides of the equation.

$x = 7$

Step 7.3

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

$(- \infty, 7) \cup (7, \infty)$

Step 8

Use each root to create test intervals.

$x < 2$

$2 < x < 7$

$x > 7$

Step 9

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 9.1

Test a value on the interval $x < 2$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < 2$ and see if this value makes the original inequality true.

$x = 0$

Step 9.1.2

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

$\frac{4 (0) - 8}{(0) - 7} \geq 0$

Step 9.1.3

The left side $1.\overline{142857}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $2 < x < 7$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $2 < x < 7$ and see if this value makes the original inequality true.

$x = 4$

Step 9.2.2

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

$\frac{4 (4) - 8}{(4) - 7} \geq 0$

Step 9.2.3

The left side $- 2.\overline{6}$ is less than the right side $0$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $x > 7$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 9.3.2

Replace $x$ with $10$ in the original inequality.

$\frac{4 (10) - 8}{(10) - 7} \geq 0$

Step 9.3.3

The left side $10.\overline{6}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.4

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

$x < 2$ True

$2 < x < 7$ False

$x > 7$ True

$x < 2$ True

$2 < x < 7$ False

$x > 7$ True

Step 10

The solution consists of all of the true intervals.

$x \leq 2$ or $x > 7$

Step 11

Convert the inequality to interval notation.

$(- \infty, 2] \cup (7, \infty)$

Step 12