Precalculus Examples

Find the Domain (5x)/( square root of 14-x)
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
Subtract from both sides of the inequality.
Step 2.2
Divide each term in by and simplify.
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Step 2.2.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 2.2.2
Simplify the left side.
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Step 2.2.2.1
Dividing two negative values results in a positive value.
Step 2.2.2.2
Divide by .
Step 2.2.3
Simplify the right side.
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Step 2.2.3.1
Divide by .
Step 3
Set the denominator in equal to to find where the expression is undefined.
Step 4
Solve for .
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Step 4.1
To remove the radical on the left side of the equation, square both sides of the equation.
Step 4.2
Simplify each side of the equation.
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Step 4.2.1
Use to rewrite as .
Step 4.2.2
Simplify the left side.
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Step 4.2.2.1
Simplify .
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Step 4.2.2.1.1
Multiply the exponents in .
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Step 4.2.2.1.1.1
Apply the power rule and multiply exponents, .
Step 4.2.2.1.1.2
Cancel the common factor of .
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Step 4.2.2.1.1.2.1
Cancel the common factor.
Step 4.2.2.1.1.2.2
Rewrite the expression.
Step 4.2.2.1.2
Simplify.
Step 4.2.3
Simplify the right side.
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Step 4.2.3.1
Raising to any positive power yields .
Step 4.3
Solve for .
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Step 4.3.1
Subtract from both sides of the equation.
Step 4.3.2
Divide each term in by and simplify.
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Step 4.3.2.1
Divide each term in by .
Step 4.3.2.2
Simplify the left side.
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Step 4.3.2.2.1
Dividing two negative values results in a positive value.
Step 4.3.2.2.2
Divide by .
Step 4.3.2.3
Simplify the right side.
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Step 4.3.2.3.1
Divide by .
Step 5
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 6