Practice Makes Perfect Algebra - Carolyn Wheater [27]
If the fractions have different denominators, as in ,
• Factor the denominators:
• Identify any factors common to both denominators, in this case, x − 2
• Form the lowest common denominator(LCD) from the product of each factor that is common, used once, and any remaining factors of either denominator, for this problem(x − 2)(x + 2)(x + 3)
• Transform each fraction by multiplying the numerator and denominator by the same quantity(don’t worry about multiplying out the denominator yet)
When the fractions have common denominators, add or subtract the numerators. For subtraction, use parentheses around the second numerator to avoid sign errors.
Finally, factor the numerator if possible, and simplify if possible. Multiply out the denominator at the very last.
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EXERCISE 11.4
Add or subtract as indicated, and give answers in simplest form.
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Complex fractions
A complex fraction is one that contains fractions within its numerator or denominator or both. There may be one fraction or several fractions in the numerator or denominator or both.
There are two methods for simplifying complex fractions. The first is to focus your attention initially on the numerator and simplify it as completely as possible and then turn to the denominator and simplify that. The final step in this method is to realize that a fraction is actually a division problem, and divide the numerator by the denominator.
To simplify the complex fraction by this method, first do the subtraction in the numerator: . Next, turn your attention to the denominator and do that addition: . Finally, divide the simplified numerator by the simplified denominator.
The second method for simplifying complex fractions is often quicker. Find the LCD for all the fractions contained in the numerator and denominator, and then multiply both the numerator and the denominator by this LCD. For the complex fraction , the LCD would be 2xy2. Multiply the complex fraction by , distribute, and simplify.
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EXERCISE 11.5
Simplify each complex fraction.
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Solving rational equations
There are two methods for solving rational equations. One uses a property of proportions and the other depends on multiplying through to clear fractions.
Cross multiplying
In the cross-multiplication method, you simplify the equation until it is two equal fractions, and then you cross multiply. Cross multiplying gives you an equation that you can solve to find the value of the variable. This method works best when the equation is relatively simple. If you use cross multiplying on rational equations with higher-degree polynomials or just a lot of polynomials in the numerators and denominators, you can end up with an equation that’s really difficult to solve, and probably extraneous solutions.
• First, concentrate on the left side of the equation. Perform all indicated operations until the left side is a single fraction.
• Next concentrate on the right side of the equation. Perform all indicated operations until the right side is a single fraction.
• Cross multiply and solve the resulting equation.
Always check your solutions in the original equation. Extraneous solutions are not unusual, especially since any values that would make one of the denominators equal to 0 are not in the domain of the equation and therefore can’t be solutions.
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EXERCISE 11.6
Solve each of the rational equations by cross multiplying.
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Multiplying through by the LCD
The second method for solving rational equations involves clearing, or eliminating, the fractions as quickly as possible. In this method, you multiply every term in the equation by a common denominator to eliminate all the fractions. In the rational equation , multiplying each term by x will give you 1 + 3x = 7, a simple