Simplify Rational Expressions
Simplify each rational expression.
- ((x^2-9)/(x+3))
- ((x^3-27)/(x-3))
- ((x^4-81)/(x^2+9))
- ((x^6-64)/(x^3-4))
- ((x^3-y^3)/(x-y))
Multiply and Divide Rational Expressions
Multiply or divide each rational expression.
- ((x+2)/(x-3)\cdot(x-5)/(x+1))
- ((x^2-9)/(x+3)\div(x-3)/(x^2-9))
- ((2x+1)/(x^2-1)\div(x-1)/(x+1))
- ((x^3-8)/(x^2+2x+4)\cdot(x^2-4)/(x^2-2x))
- ((x^2-y^2)/(x+y)\div(x-y)/(x+y)^2)
Add and Subtract Rational Expressions
Add or subtract each rational expression.
- ((x+2)/(x-3)+(x-5)/(x+1))
- ((x^2-9)/(x+3)-(x-3)/(x^2-9))
- ((2x+1)/(x^2-1)-(x-1)/(x+1))
- ((x^3-8)/(x^2+2x+4)+(x^2-4)/(x^2-2x))
- ((x^2-y^2)/(x+y)-(x-y)/(x+y)^2)
Solve Rational Equations
Solve each rational equation.
- ((x+2)/(x-3)=2)
- ((x^2-9)/(x+3)=x-3)
- ((2x+1)/(x^2-1)=x-1)
- ((x^3-8)/(x^2+2x+4)=x^2-4)
- ((x^2-y^2)/(x+y)=(x-y))
Applications of Rational Expressions
Solve each application problem.
- A rectangular garden has a length that is 3 meters more than its width. If the area of the garden is 20 square meters, find the dimensions of the garden.
- A train travels from Chicago to St. Louis, a distance of 300 miles. The train’s speed on the return trip is 10 mph faster than its speed on the trip to St. Louis. If the total travel time is 10 hours, find the train’s speed on the trip to St. Louis.
- A farmer has 120 feet of fencing to enclose a rectangular plot of land. If the length of the plot is 10 feet more than its width, find the dimensions of the plot that enclose the maximum area.
- A company produces and sells a product for $10 per unit. The company’s fixed costs are $5,000 per week, and its variable costs are $3 per unit.
(a) Find the company’s profit function.
(b) How many units must the company sell to break even?
(c) Find the company’s maximum profit.
25. A car rental company charges a daily rental fee of $50 and an additional $0.25 per mile driven. If a customer rents a car for 3 days and drives 200 miles, find the total cost of the rental.
Tips and Tricks
- Factor the numerator and denominator of each rational expression to simplify.
- Use the rules for multiplying and dividing fractions to combine rational expressions.
- Combine like terms when adding or subtracting rational expressions.
- Check your solutions by substituting them back into the original equation.
