5.6 Solve Rational Equations

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Solving Rational Equations

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5.6 Solve Rational Equations p. 345 What is the easiest way to solve a rational equation? What is an extraneous solution? How do you know when to cross multiply or multiply out the denominator?

Beware of extraneous solutions! Cross Multiplying When there is only one fraction on each side of the =, just cross multiply as if you are solving a proportion. Beware of extraneous solutions!

Write original equation. 3 x + 1 = 9 4x + 1 Solve: 3 x + 1 = 9 4x + 1 Write original equation. 3(4x + 5) = 9(x + 1) Cross multiply. 12x + 15 = 9x + 9 Distributive property 3x + 15 = 9 Subtract 9x from each side. 3x = – 6 Subtract 15 from each side. x = – 2 Divide each side by 3. The solution is –2. Check this in the original equation. ANSWER

Example: Solve. Check your solutions!

Solve Check your solutions!

ALLOYS An alloy is formed by mixing two or more metals. Sterling silver is an alloy composed of 92.5% silver and 7.5% copper by weight. Jewelry silver is composed of 80% silver and 20% copper by weight. How much pure silver should you mix with 15 ounces of jewelry silver to make sterling silver? SOLUTION = Weight of copper in mixture Total weight of mixture Percent of copper in mixture Page 345

7.5 100 0.2(15) 15 + x = x is the amount of silver added. 7.5(15 + x) = 100(0.2)(15) Cross multiply. 112.5 + 7.5x = 300 Simplify. 7.5x = 187.5 Subtract 112.5 from each side. x = 25 Divide each side by 7.5. You should mix 25 ounces of pure silver with the jewelry silver. ANSWER

Hint for solving: A quick way to solve a rational equation is to multiply everything through by the LCD. This will get rid of all the fractions! Beware of extraneous solutions!

Multiply each fraction through by the LCD Example: Solve. LCD: 2x Multiply each fraction through by the LCD Check your solution!

( ) The solutions are 1 and 15. Check these in the original equation. 1 – 8 x – 5 = 3 x Solve: 1 – 8 x – 5 = 3 x Write original equation. x(x – 5) 1– 8 x – 5 ( ) = 3 x Multiply each side by the LCD, x(x–5). x(x –5) – 8x = 3(x – 5) Simplify. x2 – 5x – 8x = 3x – 15 Simplify. x2 – 16x +15 = 0 Write in standard form. (x – 1)(x – 15) = 0 Factor. x = 1 or x = 15 Zero product property ANSWER The solutions are 1 and 15. Check these in the original equation.

Solve. LCD: ? LCD: (x+1) ? Check your solution! No Solution!

Solve. Factor 1st! LCD: (x+2)(x-2) Check your solutions!

What is the easiest way to solve a rational equation? Multiply the equation by a value that will make the denominator equal to 1. What is an extraneous solution? A solution that does not make a true statement in the original equation. How do you know when to cross multiply or multiply out the denominator? Cross multiply if there is only one fraction on each side of the equal sign, otherwise multiply out the denominator.

5.6 Assignment p. 348 5-27 odd