Trigonometry Examples

Convert to Interval Notation x-8/x<2
Step 1
Find the LCD of the terms in the equation.
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Step 1.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 1.2
The LCM of one and any expression is the expression.
Step 2
Multiply each term in by to eliminate the fractions.
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Step 2.1
Multiply each term in by .
Step 2.2
Simplify the left side.
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Step 2.2.1
Simplify each term.
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Step 2.2.1.1
Multiply by .
Step 2.2.1.2
Cancel the common factor of .
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Step 2.2.1.2.1
Move the leading negative in into the numerator.
Step 2.2.1.2.2
Cancel the common factor.
Step 2.2.1.2.3
Rewrite the expression.
Step 3
Solve the inequality.
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Step 3.1
Subtract from both sides of the inequality.
Step 3.2
Convert the inequality to an equation.
Step 3.3
Factor using the AC method.
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Step 3.3.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 3.3.2
Write the factored form using these integers.
Step 3.4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.5
Set equal to and solve for .
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Step 3.5.1
Set equal to .
Step 3.5.2
Add to both sides of the equation.
Step 3.6
Set equal to and solve for .
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Step 3.6.1
Set equal to .
Step 3.6.2
Subtract from both sides of the equation.
Step 3.7
The final solution is all the values that make true.
Step 4
Find the domain of .
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Step 4.1
Set the denominator in equal to to find where the expression is undefined.
Step 4.2
The domain is all values of that make the expression defined.
Step 5
Use each root to create test intervals.
Step 6
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 6.1
Test a value on the interval to see if it makes the inequality true.
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Step 6.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 6.1.2
Replace with in the original inequality.
Step 6.1.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 6.2
Test a value on the interval to see if it makes the inequality true.
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Step 6.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 6.2.2
Replace with in the original inequality.
Step 6.2.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 6.3
Test a value on the interval to see if it makes the inequality true.
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Step 6.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 6.3.2
Replace with in the original inequality.
Step 6.3.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 6.4
Test a value on the interval to see if it makes the inequality true.
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Step 6.4.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 6.4.2
Replace with in the original inequality.
Step 6.4.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 6.5
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
False
True
False
True
False
Step 7
The solution consists of all of the true intervals.
or
Step 8
Convert the inequality to interval notation.
Step 9