Solving Rational Equations & Word Problems

Front

What is a rational equation?

Back

A rational equation is an equation that involves at least one fraction whose numerator and/or denominator is a polynomial.

Front

How do you solve a rational equation?

Back

To solve a rational equation, find a common denominator, eliminate the fractions by multiplying both sides by that denominator, and then solve the resulting polynomial equation.

Front

What is the formula for work rate when two people work together?

Back

If person A can complete a job in time A and person B can complete it in time B, their combined work rate is given by: \( \frac{1}{A} + \frac{1}{B} = \frac{1}{T} \), where T is the time taken to complete the job together.

Front

If Jerry works twice as fast as Kenny, how can we express their work rates?

Back

Let Kenny's work rate be \( x \). Then Jerry's work rate is \( 2x \). Their combined work rate when working together is \( x + 2x = 3x \).

Front

How do you find the time taken for two workers to complete a job together?

Back

Use the formula: \( T = \frac{1}{\frac{1}{A} + \frac{1}{B}} \), where A and B are the times taken by each worker to complete the job alone.

Front

What is the significance of the least common denominator (LCD) in rational equations?

Back

The least common denominator (LCD) is used to eliminate fractions in rational equations, making it easier to solve for the variable.

Front

How do you check your solution in a rational equation?

Back

Substitute the solution back into the original equation to verify that both sides are equal, ensuring that no extraneous solutions were introduced.

Front

What is an extraneous solution?

Back

An extraneous solution is a solution that emerges from the process of solving an equation but does not satisfy the original equation.

Front

How can you express the work done by a crew that works for a certain number of days?

Back

If a crew can complete a job in D days, the work done in T days is given by: \( \text{Work} = \frac{T}{D} \).

Front

What is the formula for the combined work rate of two crews?

Back

If Crew 1 can complete a job in D1 days and Crew 2 can complete it in D2 days, their combined work rate is: \( \frac{1}{D1} + \frac{1}{D2} \).