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More Linear Equations Solving Rational Equations Example 1 3( x 4) 5( x 6) 2( x 1) 3x 12 5x 30 2x 2 3x 12 7x 32 3x 3x 12 4x 32 32 32 44 4x 11 x Before working this problem, get rid of the parentheses by distributing the outside numbers. If there are like terms on the same side of the equals sign, just combine them as they are. Solving Rational Equations Example 2 2( x 1) 3( x 2) 4( x 3) 2x 2 3x 6 4x 12 2x 2 7 x 6 2x 2x 2 5x 6 6 6 8 5x 8 x 5 Before working this problem, get rid of the parentheses by distributing the outside numbers. If there are like terms on the same side of the equals sign, just combine them as they are. Solving Rational Equations Example 3 4( x 2) 5( x 3) 2( x 1) 4x 8 5x 15 2x 2 4x 8 3x 17 3x 3x x 8 17 8 8 x 25 Before working this problem, get rid of the parentheses by distributing the outside numbers. If there are like terms on the same side of the equals sign, just combine them as they are. Solving Rational Equations Example 4 3( x 1) 2( x 5) 5( x 1) 2( x 3) 3x 3 2x 10 5x 5 2x 6 5x 7 3x 11 3x 3x 2x 7 11 7 7 2x 4 x2 Don’t forget to multiply by the number and the sign in front of the number. Solving Rational Equations Example 5 x 3( x 5) 4( x 3) ( x 4) Before working this problem, get rid of x 3x 15 4x 12 x 4 the parentheses by 2x 15 3x 8 2x 2x 15 5x 8 8 8 23 5x 23 x 5 distributing the outside numbers. If there are like terms on the same side of the equals sign, just combine them as they are. Solving Rational Equations Example 6 x 3 x 1 3 4 4 2 3 8 6 4 3 20 12 5 4 9 9 4 5 12 6 1 8 4 Fractions!!! 10 2 2 14 7 11 11 You only need to know one “trick” to solve equations with fractions. To multiply a fraction times a whole number, divide the bottom into the whole number, then multiply what’s left. Let’s practice a couple of those before we work this equation. Solving Rational Equations Example 6 x 3 x 1 12 ( ) 3 4 4 2 4x 9 3x 6 3x 3x x 9 6 9 9 x 15 Back to the problem. Find the common denominator of all of the fractions and multiply everything by that. Remember, to multiply fractions times whole numbers, divide the bottom into the whole number, then multiply what’s left. Solving Rational Equations Example 7 x 3 x 3 10 ( ) 2 5 5 2 5x 6 2x 15 2x 2x 3x 6 15 6 6 3x 9 x 3 Find the common denominator of all of the fractions and multiply everything by that. Remember, to multiply fractions times whole numbers, divide the bottom into the whole number, then multiply what’s left. Solving Rational Equations Example 8 x 3 x 15 ( 2 ) 3 5 5 5x 9 30 3x 3x 3x 8x 9 30 9 9 8x 21 21 x 8 Remember, to multiply fractions times whole numbers, divide the bottom into the whole number, then multiply what’s left. Solving Rational Equations Example 10 3( x 4) 3( x 6) 3x 12 3x 18 3x 3x 12 18 O What happened that is different from the other problems? If the variables go away, then the answer is either “no solutions” or “all real numbers”. It’s “no solutions” if what’s leftover is a false statement. It’s “all real numbers” if what’s leftover is a true statement. Solving Rational Equations Example 11 2(3 x 6) 3(2 x 4) 6x 12 6x 12 6x 6x 12 12 ALL REAL NUMBERS Remember, if the variables go away, then the answer is either “no solutions” or “all real numbers”. It’s “no solutions” if what’s leftover is a false statement. It’s “all real numbers” if what’s leftover is a true statement. Solving Rational Equations Example 12 5( x 2) 2( x 5) 5x 10 2x 10 2x 2x 3x 10 10 10 10 3x 0 3 3 x0 Is this “no solutions” or “all real numbers”? Neither. Remember, it’s only one of those if the VARIABLES go away. Just keep on working this problem. Solving Rational Equations Pair Practice: Solve these problems with a partner. 1) 2) 3) 4) 5) 5x 10 25 3x 4 5x 12 Answers : 1)3 2) 8 3)7 9 4) 8 5) 4( x 3) 2( x 1) 3( x 2) 2( x 4) x 3 4( x 1) 5(2 x 3) 2(5 x 1) x 3 x 2 6) 3 4 2 3 17 6) 2 Solving Rational Equations Answers to worksheet #1: 1) 3 2) 36 3) 17 4) 12 8 5) 5 6) 4 13 7) 4 3 8) 9)3 10) 14 11) 3 12) 13) 32 14) 3 1 15) 4 234 16) 25 The Rules are very simple. Be the first person to identify the mistake made in the following problems, raise your hand and hollar “Hey, Stupid!” The first correct answer wins you five bonus points on the Chapter Test. Once you win once, you cannot win again until next chapter. It’s time for… HEY STUPID!!! Solving Rational Equations HEY STUPID!!! #1 5( x 1) 3( x 3) 5x 5 3x 3 3x 3x 2x 5 3 5 5 2x 8 2 2 x 4 Didn’t distribute the 3 in the first step. Solving Rational Equations HEY STUPID!!! #2 2x 5 3 3 3 2x 2 2 2 x 1 Supposed to subtract the 5 instead of the 3. Solving Rational Equations HEY STUPID!!! #3 5(3x 1) 3(5 x 3) 15x 5 15x 9 15x 15x 5 9 ALL REAL NUMBERS Should be no solutions, because the statement that is leftover is NOT true. Solving Rational Equations HEY STUPID!!! #4 4( x 1) 3( x 3) 2( x 1) 4x 4 3x 9 2x 2 4x 4 x 11 Didn’t distribute the x x negative in the first step. 3x 4 11 4 4 3x 15 3 3 x 5 Solving Inequalities Hey, Stupid! #5 9 3x 21 9 9 3x 12 3 3 x 4 5 4 3 Anytime you multiply or divide by a negative, you have to turn the inequality symbol around. Solving Rational Equations Hey, Stupid! #6 x 3 x 6 ( 4 ) 3 2 2 2 x 9 4 3x 3x 3x 5x 9 4 9 9 5x 5 x 1 You have to multiply EVERYTHING by the common denominator. They didn’t multiply the four by six. Solving Inequalities Hey, Stupid! #7 3x 2 4x 8 3x 3x 2 x 8 8 8 10 x 9 10 11 When you have the x on the left, you need to rewrite the WHOLE problem by turning it all around before graphing it. You should be graphing : x 10