Homework 1 - Garcia Ilse Homework 1 Due Sep 7 2007 3:00 am...

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Garcia, Ilse – Homework 1 – Due: Sep 7 2007, 3:00 am – Inst: Fonken 1 This print-out should have 21 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. Yes, Homework 1 is due AFTER Homework 2. 001 (part 1 of 1) 10 points Rationalize the numerator of x + 4 - x - 1 x . 1. 5 x ( x + 4 - x - 1) 2. 5 x x + 4 - x - 1 3. 3 x x + 4 + x - 1 4. 3 x ( x + 4 + x - 1) 5. 5 x ( x + 4 + x - 1) correct Explanation: By the difference of squares, ( x + 4 - x - 1)( x + 4 + x - 1) = ( x + 4) 2 - ( x - 1) 2 = 5 . Thus, after multiplying both the numerator and the denominator in the given expression by x + 4 + x - 1 , we obtain 5 x ( x + 4 + x - 1) . keywords: rationalization numerator, 002 (part 1 of 1) 10 points Simplify the expression f ( x ) = 5 + 15 x - 2 2 - 6 x x 2 - 4 · as much as possible. 1. f ( x ) = x + 2 x + 4 2. f ( x ) = 5 2 x - 2 x - 4 · 3. f ( x ) = 5 2 x + 2 x - 4 · correct 4. f ( x ) = 5 2 x + 2 2 x - 4 · 5. f ( x ) = x - 2 x + 4 6. f ( x ) = x - 2 2 x + 4 Explanation: After bringing the numerator to a common denominator it becomes 5 x - 10 + 15 x - 2 = 5 x + 5 x - 2 . Similarly, after bringing the denominator to a common denominator and factoring it be- comes 2 x 2 - 8 - 6 x x 2 - 4 = 2( x + 1)( x - 4) x 2 - 4 . Consequently, f ( x ) = 5 + 15 x - 2 2 - 6 x x 2 - 4 · = 5 x + 5 2( x + 1)( x - 4) x 2 - 4 x - 2 · . On the other hand, x 2 - 4 = ( x + 2)( x - 2) . Thus, finally, we see that f ( x ) = 5 2 x + 2 x - 4 .
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Garcia, Ilse – Homework 1 – Due: Sep 7 2007, 3:00 am – Inst: Fonken 2 keywords: 003 (part 1 of 1) 10 points Find the solution set of the inequality x - 3 x - 5 < x + 6 x + 5 . 1. -∞ , - 15 i [ - 5 , 5 · 2. - 15 , - 5 · [ 5 , · 3. -∞ , - 15 · [ h - 5 , 5 · 4. h - 15 , - 5 · [ 5 , · 5. -∞ , - 15 · [ - 5 , 5 · correct Explanation: To begin we need to arrange that all the terms are on one side of the inequality. Thus the inequality becomes x - 3 x - 5 - x + 6 x + 5 < 0 , which in turn becomes ( ) x + 15 ( x + 5)( x - 5) < 0 after the right hand side is brought to a com- mon denominator. Now the right hand side changes sign at the zeros of its numerator and denominator, i.e. , at x = - 15 , 5 , - 5, and the sign chart - 15 - 5 5 - + - + determines which sign it takes in a given in- terval. Thus the solution set of ( ) is the union -∞ , - 15 · [ - 5 , 5 · . keywords: 004 (part 1 of 1) 10 points The straight line is parallel to y + 5 x = 3 and passes through the point P (5 , 4). Find its x -intercept. 1. x -intercept = 29 5 correct 2. x -intercept = - 21 5 3. x -intercept = 29 6 4. x -intercept = - 29 5 5. x -intercept = 6 Explanation: Since is parallel to the line y + 5 x = 3, these lines have the same slope - 5, Thus by the point-slope formula the equation of is given by y - 4 = - 5( x - 5) . Now the x -intercept of is the solution of y = 0. Consequently, x -intercept = 29 5 . keywords: 005 (part 1 of 1) 10 points Determine the radius of the circle given by the equation x 2 - 4 x + y 2 - 21 = 0 . 1. radius = 5 correct 2. radius = 2
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Garcia, Ilse – Homework 1 – Due: Sep 7 2007, 3:00 am – Inst: Fonken 3 3. radius = 21 4. radius = 21 5. radius = 4 6. radius = 25 Explanation: By completing the square, we see that x 2 - 4 x + y 2 - 21 = x 2 - 4 x + 4 + y 2 - 21 - 4 = ( x - 2) 2 + y 2 - 25 = 0 .
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