Finite Math Examples

Prove that a Root is on the Interval (5,6) , x+6y=5
,
Step 1
Solve the equation for in terms of .
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Step 1.1
Subtract from both sides of the equation.
Step 1.2
Divide each term in by and simplify.
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Step 1.2.1
Divide each term in by .
Step 1.2.2
Simplify the left side.
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Step 1.2.2.1
Cancel the common factor of .
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Step 1.2.2.1.1
Cancel the common factor.
Step 1.2.2.1.2
Divide by .
Step 1.2.3
Simplify the right side.
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Step 1.2.3.1
Move the negative in front of the fraction.
Step 2
The Intermediate Value Theorem states that, if is a real-valued continuous function on the interval , and is a number between and , then there is a contained in the interval such that .
Step 3
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
Step 4
Calculate .
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Step 4.1
Combine the numerators over the common denominator.
Step 4.2
Simplify the expression.
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Step 4.2.1
Subtract from .
Step 4.2.2
Divide by .
Step 5
Calculate .
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Step 5.1
Combine the numerators over the common denominator.
Step 5.2
Simplify the expression.
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Step 5.2.1
Subtract from .
Step 5.2.2
Move the negative in front of the fraction.
Step 6
Since is on the interval , solve the equation for at the root by setting to in .
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Step 6.1
Rewrite the equation as .
Step 6.2
Subtract from both sides of the equation.
Step 6.3
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 6.4
Divide each term in by and simplify.
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Step 6.4.1
Divide each term in by .
Step 6.4.2
Simplify the left side.
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Step 6.4.2.1
Dividing two negative values results in a positive value.
Step 6.4.2.2
Divide by .
Step 6.4.3
Simplify the right side.
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Step 6.4.3.1
Divide by .
Step 7
The Intermediate Value Theorem states that there is a root on the interval because is a continuous function on .
The roots on the interval are located at .
Step 8