Linear Algebra Examples

Find the Domain x+5 square root of x^2+11x+3
Step 1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 2
Solve for .
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Step 2.1
Convert the inequality to an equation.
Step 2.2
Use the quadratic formula to find the solutions.
Step 2.3
Substitute the values , , and into the quadratic formula and solve for .
Step 2.4
Simplify.
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Step 2.4.1
Simplify the numerator.
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Step 2.4.1.1
Raise to the power of .
Step 2.4.1.2
Multiply .
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Step 2.4.1.2.1
Multiply by .
Step 2.4.1.2.2
Multiply by .
Step 2.4.1.3
Subtract from .
Step 2.4.2
Multiply by .
Step 2.5
Simplify the expression to solve for the portion of the .
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Step 2.5.1
Simplify the numerator.
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Step 2.5.1.1
Raise to the power of .
Step 2.5.1.2
Multiply .
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Step 2.5.1.2.1
Multiply by .
Step 2.5.1.2.2
Multiply by .
Step 2.5.1.3
Subtract from .
Step 2.5.2
Multiply by .
Step 2.5.3
Change the to .
Step 2.5.4
Rewrite as .
Step 2.5.5
Factor out of .
Step 2.5.6
Factor out of .
Step 2.5.7
Move the negative in front of the fraction.
Step 2.6
Simplify the expression to solve for the portion of the .
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Step 2.6.1
Simplify the numerator.
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Step 2.6.1.1
Raise to the power of .
Step 2.6.1.2
Multiply .
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Step 2.6.1.2.1
Multiply by .
Step 2.6.1.2.2
Multiply by .
Step 2.6.1.3
Subtract from .
Step 2.6.2
Multiply by .
Step 2.6.3
Change the to .
Step 2.6.4
Rewrite as .
Step 2.6.5
Factor out of .
Step 2.6.6
Factor out of .
Step 2.6.7
Move the negative in front of the fraction.
Step 2.7
Consolidate the solutions.
Step 2.8
Use each root to create test intervals.
Step 2.9
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 2.9.1
Test a value on the interval to see if it makes the inequality true.
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Step 2.9.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.9.1.2
Replace with in the original inequality.
Step 2.9.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 2.9.2
Test a value on the interval to see if it makes the inequality true.
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Step 2.9.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.9.2.2
Replace with in the original inequality.
Step 2.9.2.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 2.9.3
Test a value on the interval to see if it makes the inequality true.
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Step 2.9.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.9.3.2
Replace with in the original inequality.
Step 2.9.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 2.9.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 2.10
The solution consists of all of the true intervals.
or
or
Step 3
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 4