Linear Algebra Examples

Find the Domain log base 2 of cube root of (2+x)/(4x)
Step 1
Set the argument in greater than to find where the expression is defined.
Step 2
Solve for .
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Step 2.1
To remove the radical on the left side of the inequality, cube both sides of the inequality.
Step 2.2
Simplify each side of the inequality.
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Step 2.2.1
Use to rewrite as .
Step 2.2.2
Simplify the left side.
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Step 2.2.2.1
Simplify .
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Step 2.2.2.1.1
Multiply the exponents in .
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Step 2.2.2.1.1.1
Apply the power rule and multiply exponents, .
Step 2.2.2.1.1.2
Cancel the common factor of .
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Step 2.2.2.1.1.2.1
Cancel the common factor.
Step 2.2.2.1.1.2.2
Rewrite the expression.
Step 2.2.2.1.2
Simplify.
Step 2.2.3
Simplify the right side.
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Step 2.2.3.1
Raising to any positive power yields .
Step 2.3
Solve for .
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Step 2.3.1
Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.
Step 2.3.2
Divide each term in by and simplify.
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Step 2.3.2.1
Divide each term in by .
Step 2.3.2.2
Simplify the left side.
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Step 2.3.2.2.1
Cancel the common factor of .
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Step 2.3.2.2.1.1
Cancel the common factor.
Step 2.3.2.2.1.2
Divide by .
Step 2.3.2.3
Simplify the right side.
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Step 2.3.2.3.1
Divide by .
Step 2.3.3
Subtract from both sides of the equation.
Step 2.3.4
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
Step 2.3.5
Consolidate the solutions.
Step 2.4
Find the domain of .
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Step 2.4.1
Set the denominator in equal to to find where the expression is undefined.
Step 2.4.2
Divide each term in by and simplify.
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Step 2.4.2.1
Divide each term in by .
Step 2.4.2.2
Simplify the left side.
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Step 2.4.2.2.1
Cancel the common factor of .
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Step 2.4.2.2.1.1
Cancel the common factor.
Step 2.4.2.2.1.2
Divide by .
Step 2.4.2.3
Simplify the right side.
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Step 2.4.2.3.1
Divide by .
Step 2.4.3
The domain is all values of that make the expression defined.
Step 2.5
Use each root to create test intervals.
Step 2.6
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 2.6.1
Test a value on the interval to see if it makes the inequality true.
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Step 2.6.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.6.1.2
Replace with in the original inequality.
Step 2.6.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 2.6.2
Test a value on the interval to see if it makes the inequality true.
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Step 2.6.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.6.2.2
Replace with in the original inequality.
Step 2.6.2.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 2.6.3
Test a value on the interval to see if it makes the inequality true.
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Step 2.6.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.6.3.2
Replace with in the original inequality.
Step 2.6.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 2.6.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 2.7
The solution consists of all of the true intervals.
or
or
Step 3
Set the denominator in equal to to find where the expression is undefined.
Step 4
Divide each term in by and simplify.
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Step 4.1
Divide each term in by .
Step 4.2
Simplify the left side.
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Step 4.2.1
Cancel the common factor of .
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Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.3
Simplify the right side.
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Step 4.3.1
Divide by .
Step 5
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 6