Trigonometry Examples

Solve for x log base 8 of x^2-x> log base 8 of 42
Step 1
Convert the inequality to an equality.
Step 2
Solve the equation.
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Step 2.1
For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.
Step 2.2
Solve for .
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Step 2.2.1
Subtract from both sides of the equation.
Step 2.2.2
Factor using the AC method.
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Step 2.2.2.1
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 2.2.2.2
Write the factored form using these integers.
Step 2.2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.2.4
Set equal to and solve for .
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Step 2.2.4.1
Set equal to .
Step 2.2.4.2
Add to both sides of the equation.
Step 2.2.5
Set equal to and solve for .
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Step 2.2.5.1
Set equal to .
Step 2.2.5.2
Subtract from both sides of the equation.
Step 2.2.6
The final solution is all the values that make true.
Step 3
Find the domain of .
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Step 3.1
Set the argument in greater than to find where the expression is defined.
Step 3.2
Solve for .
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Step 3.2.1
Multiply both sides by .
Step 3.2.2
Simplify.
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Step 3.2.2.1
Simplify the left side.
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Step 3.2.2.1.1
Simplify .
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Step 3.2.2.1.1.1
Simplify terms.
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Step 3.2.2.1.1.1.1
Cancel the common factor of .
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Step 3.2.2.1.1.1.1.1
Cancel the common factor.
Step 3.2.2.1.1.1.1.2
Rewrite the expression.
Step 3.2.2.1.1.1.2
Apply the distributive property.
Step 3.2.2.1.1.1.3
Simplify the expression.
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Step 3.2.2.1.1.1.3.1
Multiply by .
Step 3.2.2.1.1.1.3.2
Move to the left of .
Step 3.2.2.1.1.2
Rewrite as .
Step 3.2.2.2
Simplify the right side.
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Step 3.2.2.2.1
Multiply by .
Step 3.2.3
Solve for .
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Step 3.2.3.1
Convert the inequality to an equation.
Step 3.2.3.2
Factor out of .
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Step 3.2.3.2.1
Factor out of .
Step 3.2.3.2.2
Factor out of .
Step 3.2.3.2.3
Factor out of .
Step 3.2.3.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.2.3.4
Set equal to .
Step 3.2.3.5
Set equal to and solve for .
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Step 3.2.3.5.1
Set equal to .
Step 3.2.3.5.2
Add to both sides of the equation.
Step 3.2.3.6
The final solution is all the values that make true.
Step 3.2.4
Use each root to create test intervals.
Step 3.2.5
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 3.2.5.1
Test a value on the interval to see if it makes the inequality true.
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Step 3.2.5.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 3.2.5.1.2
Replace with in the original inequality.
Step 3.2.5.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 3.2.5.2
Test a value on the interval to see if it makes the inequality true.
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Step 3.2.5.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 3.2.5.2.2
Replace with in the original inequality.
Step 3.2.5.2.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 3.2.5.3
Test a value on the interval to see if it makes the inequality true.
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Step 3.2.5.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 3.2.5.3.2
Replace with in the original inequality.
Step 3.2.5.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 3.2.5.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 3.2.6
The solution consists of all of the true intervals.
or
or
Step 3.3
The domain is all values of that make the expression defined.
Step 4
Use each root to create test intervals.
Step 5
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 5.1
Test a value on the interval to see if it makes the inequality true.
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Step 5.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.1.2
Replace with in the original inequality.
Step 5.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 5.2
Test a value on the interval to see if it makes the inequality true.
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Step 5.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.2.2
Replace with in the original inequality.
Step 5.2.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 5.3
Test a value on the interval to see if it makes the inequality true.
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Step 5.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.3.2
Replace with in the original inequality.
Step 5.3.3
Determine if the inequality is true.
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Step 5.3.3.1
The equation cannot be solved because it is undefined.
Step 5.3.3.2
The left side has no solution, which means that the given statement is false.
False
False
False
Step 5.4
Test a value on the interval to see if it makes the inequality true.
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Step 5.4.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.4.2
Replace with in the original inequality.
Step 5.4.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 5.5
Test a value on the interval to see if it makes the inequality true.
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Step 5.5.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 5.5.2
Replace with in the original inequality.
Step 5.5.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 5.6
Compare the intervals to determine which ones satisfy the original inequality.
True
False
False
False
True
True
False
False
False
True
Step 6
The solution consists of all of the true intervals.
or
Step 7
The result can be shown in multiple forms.
Inequality Form:
Interval Notation:
Step 8