Question

$y=x+\frac{1}{x}$ at $y=\frac{17}{4} ;$ where $x>1$.

   $y=x+\frac{1}{x}$ at $y=\frac{17}{4} ;$ where $x>1$.

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Solved by Expert Taylor Shimono

08/26/2019

Step 1

We can do this by substituting $y$ into the equation and solving for $x$: \[x+\frac{1}{x}=\frac{17}{4}\] After solving this equation, we find that $x=4$. Show more…

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$y=x+\frac{1}{x}$ at $y=\frac{17}{4} ;$ where $x>1$.

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Key Concepts

Domain Restrictions

This concept deals with the limitations placed on the variable, such as x > 1 in the problem. Domain restrictions are critical in determining the validity of the solutions obtained since they help exclude extraneous solutions that don't satisfy the original conditions.

Rational Equations

This concept involves equations in which variables appear in the denominators as well as numerators. Solving such equations often requires careful manipulation to avoid undefined expressions and ensure that all potential solutions are valid within the domain.

Clearing Denominators

Clearing denominators is a common strategy used to simplify rational equations. It involves multiplying both sides of the equation by a common denominator to eliminate fractional expressions, facilitating the conversion of the original equation into a polynomial equation.

Quadratic Equations

After eliminating denominators, the resulting equation can sometimes be a quadratic equation. Understanding how to manipulate and solve quadratic equations—using methods such as factoring or the quadratic formula—is essential to finding the solution set for the variable.

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