Practice Makes Perfect Algebra - Carolyn Wheater [26]
9. The volume of a gas varies directly with temperature and inversely with pressure. At a temperature of 450°F and a pressure of 40 psi, a gas has a volume of 10 ft3. What is its volume if temperature is reduced to 440°F and pressure is raised to 50 psi?
10. The electric resistance of a wire varies directly with its length and inversely with the square of its diameter. A wire 80 ft long with a diameter of in. has a resistance of ohm. What is the resistance in a piece of the same type of wire that is 120 ft long and has a diameter of in.?
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•11•
Rational equations and their graphs
A rational expression is the quotient of two polynomials, and a rational equation is an equation containing rational expressions. Since division by 0 is impossible, the denominator of a quotient can never be 0. The domain of a rational expression is the set of all values of the variable for which the expression is defined, the values that do not make the denominator 0.
Simplifying rational expressions
The quotient of two polynomials can look very complicated, but just like fractions, rational expressions can often be simplified. When we work with fractions, we often talk about reducing the fraction to lowest terms. That wording is a little bit of a misrepresentation, however. Reducing is making something smaller, and we don’t change the value of the fraction at all. We just change the way it looks.
To simplify a rational expression, factor the numerator and the denominator and cancel any factors that appear in both. The rational expression looks like it would be difficult to work with, but if you pause for a minute and examine the numerator and denominator separately, you’ll see that both factor.
Since both the numerator and the denominator have a factor of 2x + 3, you can think of · 1, or simply
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EXERCISE 11.1
Reduce each expression to simplest form.
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Multiplying rational expressions
The basic rule for multiplying rational expressions is the same as the basic rule for fractions: numerator times numerator and denominator times denominator. Just as with fractions, however, much time and effort can be saved by “canceling,” or simplifying, before multiplying.
To multiply rational expressions:
• Factor all numerators and denominators
• Cancel any factor that appears in both a numerator and a denominator
• Multiply numerator times numerator and denominator times denominator
Multiplying might look like an unreasonable task, but begin by focusing on numerators and denominators, one by one, and factoring as completely as possible.
Once you’ve factored, you can see that there are factors that appear more than once. Cancel them out, one from a numerator with one from a denominator.
What’s left will still require some work to multiply, but it’s much easier than the original problem.
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EXERCISE 11.2
Multiply the rational expressions and give answers in simplest form.
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Dividing rational expressions
Take a minute to think about how you divide fractions. In fact, you don’t. You multiply by the reciprocal of the divisor. Use the same tactic to divide rational expressions.
To divide rational expressions, invert the divisor and multiply. Factor all the numerators and denominators and cancel where possible.
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EXERCISE 11.3
Divide the rational expressions and give the answers in simplest form.
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Adding and subtracting rational expressions
Adding and subtracting rational expressions calls on the same skills as adding and subtracting fractions. If the fractions have different denominators, they must be transformed to have a common denominator. Once the denominators are the same, you add