02_Chapter 1 - Equations and Inequalities

02_Chapter 1 - Equations and Inequalities - SECTION 1-1 31...

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SECTION 1-1 31 CHAPTER 1 Section 1-1 1. 4( x + 5) = 6( - 2) 3. 5 + 4( w - 1) = 2 + 2( + 4) 4 + 20 = 6 - 12 5 + 4 - 4 = 2 + 2 + 4 -2 = -32 4 + 1 = 4 + 4 = 16 1 = 4 Solution: 16 No solution 5. 5 - 3 a ± 4 5 = 7 ± 2 2 LCD = 10 10 · 5 – 10 · (3 ± 4) 5 = 10 · (7 ± 2 ) 2 50 - 2(3 - 4) = 5(7 - 2 ) 50 - 6 + 8 = 35 - 10 -6 + 58 = 35 - 10 4 = -23 Common Error: After line 2, students often write 50 – 2 50 ± 1 / 0 3 ± 4 / 8 = L 50 ± 6 ± 4 = L forgetting to distribute the –2. Put compound numerators in parentheses to avoid this. = - 23 4 or -5.75 7. 2 + 2 ± 1 3 = 3 + 4 4 LCD = 12 12 · 2 + 12 · (2 ± 1) 3 = 12 · + 4 6 + 4(2 - 1) = 3(3 + 4) 6 + 8 - 4 = 9 + 12 14 - 4 = 9 + 12 5 = 16 = 16 5 or 3.2 9. 0.1( t + 0.5) + 0.2 = 0.3( - 0.4) 0.1 + 0.05 + 0.2 = 0.3 - 0.12 0.3 + 0.05 = 0.3 - 0.12 0.05 = -0.12 No solution 11. 0.35( s + 0.34) + 0.15 = 0.2 - 1.66 0.35 + 0.119 + 0.15 = 0.2 - 1.66 0.5 + 0.119 = 0.2 - 1.66 0.3 = -1.779 = -5.93 Solution: -5.93 13. 2 y + 5 2 = 4 - 2 3 Excluded value: ± 0 LCD = 6 6 · 2 + 6 · 5 2 = 6 (4) - 6 · 2 3 12 + 15 = 24 - 4 -9 = -16 = 16 9 or 1.7 - 15. z ± 1 = 1 ± 1 + 2 Excluded value: z ± 1 LCD = - 1 ( - 1) ± 1 = ( - 1) 1 ± 1 + 2( - 1) = 1 + 2 - 2 - = -1 = 1 No solution: 1 is excluded 17. 2 m 5 + ± 4 6 = 4 + 1 4 - 2 LCD = 60 60 · 2 5 + 60 · ( ± 6 = 60 · (4 + 4 - 60(2) 24 + 10( - 4) = 15(4 + 1) - 120 24 + 10 - 40 = 60 + 15 - 120 34 - 40 = 60 - 105 -26 = -65 = 65 26 = 5 2 or 2.5
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32 CHAPTER 1 EQUATIONS AND INEQUALITIES 19. 1 - x ± 3 ± 2 = 2 ± 3 ± 2 Excluded value: x ± 2 LCD = – 2 ( - 2)1 - ( - 2) ( ± 3) ± 2 = ( - 2) (2 ± ± 2 - 2 - ( - 3) = 2 – 3 - 2 - + 3 = 2 – 3 1 = 2 – 3 4 = 2 2 = No solution: 2 is excluded 21. 6 y + 4 + 1 = 5 2 + 8 Excluded value: ± -4 LCD 2( + 4) 2( + 4) 6 + 4 + 2( + 4)1 = 2( + 4) 5 2( + 4) 12 + 2 + 8 = 5 2 + 20 = 5 2 = -15 = - 15 2 or -7.5 23. 3 a ± 1 2 + 4 + 4 - 3 2 + 2 = 3 3 ± 1 ( + 2) 2 - 3 ( + = 3 Excluded values: ± 0, —2 ( + 2) 2 (3 ± 1) ( + 2 - ( + 2) 2 3 ( + = ( + 2) 2 3 (3 - 1) - ( + 2)3 = ( + 2) 2 3 3 2 - - 3 - 6 = 3( 2 + 4 + 4) 3 2 - 4 - 6 = 3 2 + 12 + 12 -16 = 18 = - 9 8 or -1.125 25. 3.142 - 0.4835( - 4) = 6.795 3.142 - 0.4835 + 1.934 = 6.795 2.6585 + 1.934 = 6.795 2.6585 = 4.861 = 4.861 2.6585 = 1.83 to 3 significant digits Solution: 1.83 27. 2.32 ± 2 - 3.76 = 2.32 Excluded values: ± 0, 2 LCD = ( - 2) ( - 2) 2.32 ± 2 - ( - 2) 3.76 = 2.32 ( - 2) 2.32 2 - 3.76( - 2) = 2.32 ( - 2) 2.32 2 - 3.76 + 7.52 = 2.32 2 - 4.64 -3.76 + 7.52 = -4.64 7.52 = -0.88 = -8.55 Solution: -8.55
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SECTION 1-1 33 29. a n = 1 + ( - 1) d 1 + ( n - 1) = ( - 1) = - a 1 = ± 1 ± 1 31. 1 f = 1 1 + 1 2 LCD = 1 2 1 2 1 = 1 2 1 1 + 1 2 1 2 1 2 = 2 + 1 2 + 1 = 1 2 ( 2 + 1 ) = 1 2 = 1 2 2 + 1 33. A = 2 ab + 2 ac + 2 bc 2 + 2 + 2 = 2 + 2 = - 2 (2 b + 2 c ) = - 2 = ± 2 2 + 2 35. y = 2 x ± 3 3 + 5 (3 + 5) = 2 - 3 3 xy + 5 = 2 - 3 5 + 3 = 2 - 3 5 + 3 = (2 - 3 ) 5 + 3 2 ± 3 = = 5 + 3 2 ± 3 37. The "solution" is incorrect. Although 3 is a solution of the two last equations, they are not equivalent to the first equation because both sides have been multiplied by - 3, which is zero when = 3. It is not permitted to multiply both sides of an equation by zero. When = 3, the first equation involves division by zero. Since 3, the only possible solution, is not a solution, the given (first) equation has no solution. 39. ± 1 1 + 1 = 3 Excl. val.: ± 0 xx ± 1 () 1 + 1 = 3 2 ± 1 + 1 = 3 Excl. val.: ± -1 ( – 1) ( + 1) 1 ( ± 1) ( + + 1 = 3 1 - 1 = 3 = 4 Solution: 4 41. + 1 ± 2 1 ± 1 = + 2 Excl. val.: ± 0 + 1 ± 2 1 ± 1 = + 2 2 + ± 2 ± 1 = + 2 Excl. val.: ± 1 1 ( ± + 2) ± 1 = + 2 1 + 2 = + 2 Solution: All real numbers except the excluded numbers 0 and 1.
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34 CHAPTER 1 EQUATIONS AND INEQUALITIES 43. y = a 1 + b x + c = ( + ) ( + )1 + + () = ( + ) + + = ax + ac + + ( + + ) = + xy + cy + by = + cy + by - = ax - xy + - = ( a - y ) + ± ± = = + ± ± 45. Let = the number, Then 10 less than two thirds the number is one fourth the number 2 3 – 10 = 1 4 2 3 - 10 = 1 4 x 12 2 3 ± ² ³ ´ µ - 12(10) = 12 1 4 ± ² ³ ´ µ 8 - 120 = 3 -120 = -5 = 24 The number is 24.
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This note was uploaded on 03/31/2008 for the course MATH 1041 taught by Professor Unknown during the Spring '08 term at Temple.

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02_Chapter 1 - Equations and Inequalities - SECTION 1-1 31...

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