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Trigonometry Examples
Step 1
Convert the inequality to an equation.
Step 2
Step 2.1
Rewrite as .
Step 2.2
Let . Substitute for all occurrences of .
Step 2.3
Factor using the AC method.
Step 2.3.1
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 2.3.2
Write the factored form using these integers.
Step 2.4
Replace all occurrences of with .
Step 2.5
Rewrite as .
Step 2.6
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 2.7
Simplify.
Step 2.7.1
Move to the left of .
Step 2.7.2
Raise to the power of .
Step 2.8
Rewrite as .
Step 2.9
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 2.10
Factor.
Step 2.10.1
Simplify.
Step 2.10.1.1
Multiply by .
Step 2.10.1.2
One to any power is one.
Step 2.10.2
Remove unnecessary parentheses.
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Step 4.1
Set equal to .
Step 4.2
Add to both sides of the equation.
Step 5
Step 5.1
Set equal to .
Step 5.2
Solve for .
Step 5.2.1
Use the quadratic formula to find the solutions.
Step 5.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 5.2.3
Simplify.
Step 5.2.3.1
Simplify the numerator.
Step 5.2.3.1.1
Raise to the power of .
Step 5.2.3.1.2
Multiply .
Step 5.2.3.1.2.1
Multiply by .
Step 5.2.3.1.2.2
Multiply by .
Step 5.2.3.1.3
Subtract from .
Step 5.2.3.1.4
Rewrite as .
Step 5.2.3.1.5
Rewrite as .
Step 5.2.3.1.6
Rewrite as .
Step 5.2.3.1.7
Rewrite as .
Step 5.2.3.1.7.1
Factor out of .
Step 5.2.3.1.7.2
Rewrite as .
Step 5.2.3.1.8
Pull terms out from under the radical.
Step 5.2.3.1.9
Move to the left of .
Step 5.2.3.2
Multiply by .
Step 5.2.3.3
Simplify .
Step 5.2.4
Simplify the expression to solve for the portion of the .
Step 5.2.4.1
Simplify the numerator.
Step 5.2.4.1.1
Raise to the power of .
Step 5.2.4.1.2
Multiply .
Step 5.2.4.1.2.1
Multiply by .
Step 5.2.4.1.2.2
Multiply by .
Step 5.2.4.1.3
Subtract from .
Step 5.2.4.1.4
Rewrite as .
Step 5.2.4.1.5
Rewrite as .
Step 5.2.4.1.6
Rewrite as .
Step 5.2.4.1.7
Rewrite as .
Step 5.2.4.1.7.1
Factor out of .
Step 5.2.4.1.7.2
Rewrite as .
Step 5.2.4.1.8
Pull terms out from under the radical.
Step 5.2.4.1.9
Move to the left of .
Step 5.2.4.2
Multiply by .
Step 5.2.4.3
Simplify .
Step 5.2.4.4
Change the to .
Step 5.2.5
Simplify the expression to solve for the portion of the .
Step 5.2.5.1
Simplify the numerator.
Step 5.2.5.1.1
Raise to the power of .
Step 5.2.5.1.2
Multiply .
Step 5.2.5.1.2.1
Multiply by .
Step 5.2.5.1.2.2
Multiply by .
Step 5.2.5.1.3
Subtract from .
Step 5.2.5.1.4
Rewrite as .
Step 5.2.5.1.5
Rewrite as .
Step 5.2.5.1.6
Rewrite as .
Step 5.2.5.1.7
Rewrite as .
Step 5.2.5.1.7.1
Factor out of .
Step 5.2.5.1.7.2
Rewrite as .
Step 5.2.5.1.8
Pull terms out from under the radical.
Step 5.2.5.1.9
Move to the left of .
Step 5.2.5.2
Multiply by .
Step 5.2.5.3
Simplify .
Step 5.2.5.4
Change the to .
Step 5.2.6
The final answer is the combination of both solutions.
Step 6
Step 6.1
Set equal to .
Step 6.2
Add to both sides of the equation.
Step 7
Step 7.1
Set equal to .
Step 7.2
Solve for .
Step 7.2.1
Use the quadratic formula to find the solutions.
Step 7.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 7.2.3
Simplify.
Step 7.2.3.1
Simplify the numerator.
Step 7.2.3.1.1
One to any power is one.
Step 7.2.3.1.2
Multiply .
Step 7.2.3.1.2.1
Multiply by .
Step 7.2.3.1.2.2
Multiply by .
Step 7.2.3.1.3
Subtract from .
Step 7.2.3.1.4
Rewrite as .
Step 7.2.3.1.5
Rewrite as .
Step 7.2.3.1.6
Rewrite as .
Step 7.2.3.2
Multiply by .
Step 7.2.4
Simplify the expression to solve for the portion of the .
Step 7.2.4.1
Simplify the numerator.
Step 7.2.4.1.1
One to any power is one.
Step 7.2.4.1.2
Multiply .
Step 7.2.4.1.2.1
Multiply by .
Step 7.2.4.1.2.2
Multiply by .
Step 7.2.4.1.3
Subtract from .
Step 7.2.4.1.4
Rewrite as .
Step 7.2.4.1.5
Rewrite as .
Step 7.2.4.1.6
Rewrite as .
Step 7.2.4.2
Multiply by .
Step 7.2.4.3
Change the to .
Step 7.2.4.4
Rewrite as .
Step 7.2.4.5
Factor out of .
Step 7.2.4.6
Factor out of .
Step 7.2.4.7
Move the negative in front of the fraction.
Step 7.2.5
Simplify the expression to solve for the portion of the .
Step 7.2.5.1
Simplify the numerator.
Step 7.2.5.1.1
One to any power is one.
Step 7.2.5.1.2
Multiply .
Step 7.2.5.1.2.1
Multiply by .
Step 7.2.5.1.2.2
Multiply by .
Step 7.2.5.1.3
Subtract from .
Step 7.2.5.1.4
Rewrite as .
Step 7.2.5.1.5
Rewrite as .
Step 7.2.5.1.6
Rewrite as .
Step 7.2.5.2
Multiply by .
Step 7.2.5.3
Change the to .
Step 7.2.5.4
Rewrite as .
Step 7.2.5.5
Factor out of .
Step 7.2.5.6
Factor out of .
Step 7.2.5.7
Move the negative in front of the fraction.
Step 7.2.6
The final answer is the combination of both solutions.
Step 8
The final solution is all the values that make true.
Step 9
Use each root to create test intervals.
Step 10
Step 10.1
Test a value on the interval to see if it makes the inequality true.
Step 10.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.1.2
Replace with in the original inequality.
Step 10.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 10.2
Test a value on the interval to see if it makes the inequality true.
Step 10.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.2.2
Replace with in the original inequality.
Step 10.2.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 10.3
Test a value on the interval to see if it makes the inequality true.
Step 10.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.3.2
Replace with in the original inequality.
Step 10.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 10.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 11
The solution consists of all of the true intervals.
or
Step 12