6.7 Applications. 7.1 Rational Expressions. Answers on page 15

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1 1 6.7 Applications Answers on page The length of a CD jewel case is cm more than its width. The area of the rectangular top of the case is 168 cm². Find the length and width of the jewel case.. A 10-gallon aquarium is in. higher than it is wide. Its length is 1 in. and its volume is 70 in³. What are the height and width of the aquarium?. Find three consecutive odd integers such that times the sum of all three is 18 more than the product of the rst and second integers. 4. Find three consecutive even integers such that the sum of the squares of the rst and second integers is equal to the square of the third integer. 5. Hercomer works due north of home. Mergatroid works due east. They leave for work at the same time. By the time Hercomer is 5 mi from home, the distance between them is 1 mi more than Mergatroid's distance from home. How far from home is Mergatroid? 6. An ball is propelled from a height of 48 ft with an initial upward velocity of ft per sec. Its height h after t seconds is given by the formula h = 16t + t (a) After how many seconds is the height 64 ft? (b) After how many seconds is the height 60 ft? (c) After how many seconds does the ball hit the ground? 7.1 Rational Expressions Answers on page Find the undened values for:. Find the undened values for:. Find the undened values for: 1 5y x + 1 x 6 4x x Find the 5m + undened values for: m + m 6 5. Reduce: 6. Reduce: 4 (y ) 10 (y ) m n m + n 7. Reduce: 8. Reduce: x + x 15 x + 6x + 5 z + 7 z z + 9z 9. Which of the following are equal to 1 (more than one answer is possible): (a) x + x (b) x x (c) x + + x x + (d) x

2 Reduce: m 1 1 m 11. Multiply and reduce: 1. Multiply and reduce: 1. Multiply and reduce: 14. Divide and reduce: 15. Divide and reduce: 15a a (c + d) 18 6(c + d) x (x + ) x + 6x x x 1 x x + (x ) 6x x x 16. Simplify: 17. Simplify: 18. Simplify: 19. Simplify: 7 z 4 1 z 18 p + 4p 5 p + 7p + 10 p 1 p + 4 k k 1 k + 5k + 4k 1 k + k m + m + m + 5m + 4 m + 10m + 4 m + 5m Adding and Subtracting Rational Expressions Answers on the next page 1. Find the LCD for Find the LCD for 5p and 1 0 and 15 6p. Find the LCD for 7 6p and 15 4p 8 4. Find the LCD for 5. Find the LCD for 7 6r 1 and 5 9r 18 9 p q and 18 q p 5 6. Find the LCD for: p + 8p + 15, p p 18 and p p 0 7. Rewrite with new denominator (i.e., expand): a + b a + ab b =? a b + a b ab 8. Simplify: 4 m + 7 m 9. Simplify: 10. Simplify: 11. Simplify: 1. Simplify: 1. Simplify: 14. Simplify: 15. Simplify: 16. Simplify: 17. Simplify: 5m m m m + 1 z x p p + x + 9 x x + 8 x y 4y y 1 4p p 5 x 9 x + x + 4x + x + y x + xy + y + x y x + 4xy + y

3 15 Ÿ by 14 cm. 7, 9, (a) ; (b) ; (c). H 1 in, W 10 in 4. 6, 8, 10 or, 0, 5. 1 mi Ÿ y 0. x 6. x 5 4. m, m n x x + 1 z + z 9. (b) and (d) 10. (m + 1) 11. a c + d x + x (x + ) x (x 1) x (x ) p + 4 p xit 19. m + 6 m + Ÿ p. 1p (p ) (r ) 5. either p q or q p 6. (p + 5) (p + ) (p 6) ab (a + b) m m 1 m z x p p x + 4 x p x 6x (x + ) (x ) (x + 1) 17. x (x + y)

4 Solving Equations with Rational Expressions Answers on page Find all undened values for x: 5 x = 1. Solve:. Solve: 4. Solve: 5. Solve: 6. Solve: 5 m m = 8 z 1 4 = z + k k 4 5 = 4 k 4 a a = 4 x 1 + 4x 4 = 7 4 x + x 7. Solve: x + 5x + 6 x + x 8. Solve for F : m = kf a 9. Solve for A: h = A B + b 10. Solve for z: 9x + z = 5 y = 5x x + x 7.5 Applications of Rational Expressions Answers on page In a certain fraction, the denominator is 6 more than the numerator. If is added to both the numerator and the denominator, the resulting fraction is equivalent to 5 7. What was the original fraction?. In the 00 Winter Olympics, Catriona LeMay Doan of Canada won the 500-m skating event for women. Her rate was m per sec. What was her time (to the nearest hundredth of a second)?. A boat can go 0 mi against a current in the same time that it can go 60 mi with the current. The speed of the current is 4 mph. Find the speed of the boat in still water. 4. Working alone, Hercomer can paint a large room in 8 hr. Mergatroid can paint the same size room working alone in 6 hr. How long will it take them if they work together? 5. One pipe can ll a pool in 6 hr, and another can do it in 9 hr. How long would it take the two pipes working together to ll the pool? 6. If the water in Hercomer's bathtub is turned on full force, it can ll the tub in 4 minutes if the drain is plugged completely. The plug is leaky, though, and it will drain a full tub in 0 minutes. How long would it take to ll the tub with the water on full force with the leaky plug?

5 Variation Answers on the following page 1. If a varies directly as c, and a = 7. when c =.4, nd a when c = 5.. If z varies inversely as w, and z = 10 when w = 0.5, nd z when w = 8.. If p varies jointly as q and r, and p = 00 when q = and r =, nd p when q = 5 and r =. 4. For an object falling from rest in a vacuum, the distance the object falls varies directly as the square of the time. If an object falls 576 ft in 6 sec, how far would it fall in 4 sec? 4.1 Introduction to Variables Answers on the next page 1. Is the ordered pair (0, 8) a solution of the equation x + y = 8?. Is the ordered pair (, -1) a solution of the equation x + y = 5?. Is the ordered pair (5, ) a solution of the equation 5x y = 15? 4. Complete the ordered pair (5,?) as a solution of the equation y = x Complete the ordered pair (?, ) as a solution of the equation y = x Complete the table of values for the equation y + = 0. x y Plot the following points on a rectangular coordinate system. Label the points. (a) A = (6, ) (b) B = (-4, ) (c) C = (0, 4) (d) D = (4, 0) 6. Complete the table of values for the equation 4x 9y = 6. x y Complete the table of values for the equation x = 1. x y 8 0

6 18 Ÿ x, 0 4. (4 is extraneous) 6. x = 9. A = h (B + b). m = x = 6 or 1. z = a = 1 8. F = am k 10. z = xy 5x 9xy Ÿ sec.. 8 mph hr. (approx :6) 5..6 hr (or :6) minutes (or hr) Ÿ ft. Ÿ Yes, it is a solution.. Yes, it is a solution.. No, it is not a solution. 4. (5, 17) 5. (5, ) 6. x y x y x y graph 10. graph

7 Complete the table of values for x y = 4, then plot the ordered pairs. x 0 4 y 0 4. Graphing Linear Equations Answers on page When graphing an equation, 1. Make a table of values.. Plot the points in your table of values.. Connect the points with a line. Remember to use arrowheads, since the line extends forever in both directions. 1. Graph the equation y = x Find the x-intercept and y-intercept of: x y = 4.. Find the x-intercept and y-intercept of: 5x y = Graph the equation: x = y Graph the equation: x y = Graph the equation: x + y = Graph the equation: y x = Graph the equation: y = Graph the equation: x + = The expected height y (in centimeters) of a woman is related to the length x of her radius bone (from the elbow to the thumbside of the wrist), and is approximated by the linear equation: y =.9x Use this information for the following questions. (a) Find the expected height of a woman whose radius bone is cm long. (b) How long would the radius bone be for a woman who is 167 cm tall? 11. The expected weight y (in pounds) of a man taller than 60 in. can be approximated by the linear equation: y = 5.5x 0, where x is the height of the man in inches. Use this equation for the following questions. (a) Find the expected weight of a man whose height is 66 in. (b) How tall would a man be who weighs 155 lb?

8 0 4. Slope of a Line Answers on page 1. Use the indicated points to nd the slope for this line: points: (, 4) and (, 7). 6. Find the slope of a line containing the points: (1, ) and (1, -7). 7. Find the slope of the line y = 5x Find the slope of the line 4y = x Find the slope of the line y = 5.. Use the indicated points to nd the slope for this line: 10. Find the slope of the line x = For the line x + y = 7: (a) What is the slope of a line parallel to it? (b) What is the slope of a line perpendicular to it?. Use the indicated points to nd the slope for this line: 1. Find the slopes of these two lines and determine if they are parallel, perpendicular, x + 5y = 4 or neither: 4x + 10y = 1 1. Find the slopes of these two lines and determine if they are parallel, perpendicular, 8x 9y = 6 or neither: 8x + 6y = Find the slopes of these two lines and determine if they are parallel, perpendicular, x y = 6 or neither: x + y = 4. Find the slope of a line containing the points: (1, ) and (, 7). 15. Find the slopes of these two lines and determine if they are parallel, perpendicular, 5x y = 1 or neither: x 5y = Find the slope of a line containing the

9 1 4.4 Equations of Lines Answers on the following page 1. Determine the slope and y-intercept from this graph. Then write the equation of the line in slope-intercept form.. Find an equation of a line with slope m = 4 and with y-intercept (0, -). Give the answer in slope-intercept form. 4. Write the equation of a line with slope m = 0 and containing the point (0, ). 5. Write the equation of a line with undened slope m and containing the point (0, -). 6. Graph the line y = x + using the slope and y-intercept. 7. Graph the line passing through the point (1, -5) and having slope m = Graph the line passing through the point (-, ) and having slope m = 0.. Determine the slope and y-intercept from this graph. Then write the equation of the line in slope-intercept form. 9. Find an equation for the line containing the point (-, 5) and having slope m =. Give the answer in slope-intercept form. 10. Find an equation for the line containing the points (4, 10) and (6, 1). Give the answer in slope-intercept form. 11. Find an equation for the line containing the points (-4, 0) and (0, ). Give the answer in slope-intercept form. 1. Find an equation for the line containing the point (, -) and parallel to x = 4y+5. Give the answer in slope-intercept form. 1. Find an equation for the line perpendicular to x y = 7 and with y-intercept (0, -). Give the answer in slope-intercept form.

10 Ÿ4. 1. graph. (1, 0), (0, 8). (4, 0), (0, 10) 4. graph 5. graph 6. graph 7. graph 8. graph 9. graph 10. (a) 159 cm; (b) 4 cm 11. (a) 14 lb; (b) 68 in Ÿ neither undened undened 11. (a) ; (b) perpendicular parallel 15. neither Ÿ y = x. y = x +. y = 4x 5. x = 0 6. graph 7. graph 9. y = x y = x y = 4 x 9 1. y = x 4. y = 8. graph 11. y = 1 x +

11 4.5 Functions Answers on page 5 1. For the relation described by (1, 1), (1, -1), (0, 0), (, 4), (, -4)}: (a) Is the relation a function? (b) Find the domain: (c) Find the range: (c) Find the range: 5. For the relation shown in this graph:. For the relation described by (, 5), (, 7), (4, 9), (5, 11)}: (a) Is the relation a function? (b) Find the domain: (c) Find the range:. For the relation shown in this graph: (a) Is the relation a function? (b) Find the domain: (c) Find the range: 6. For the relation described by y = x : (a) Is the relation a function? (b) Find the domain: (c) Find the range: 4. For the relation shown in this graph: (a) Is the relation a function? (b) Find the domain: (c) Find the range: 7. For the relation described by y = x 4 : (a) Is the relation a function? (b) Find the domain: (c) Find the range: f (x) = x + 4 and g (x) = x + 4x + 1 for the following problems. 8. Find f (0) 9. Find g ( ) 10. Find f (p) 11. Find g ( x) (a) Is the relation a function? (b) Find the domain: 1. Find f (x + )

12 4 8.1 Solving Linear Systems by Graphing Answers on the following page 1. Is the ordered pair (7, ) a solution of the 4x = 6 y system: x = 9 + 4y. Is the ordered pair (6, 8) a solution of the y = x + 10 system: y = x + 0. Solve this system by graphing: x y = x + y = 6 x y = 6 y = x + 5. Solve this system by graphing: x y = 6 4x y = 8 6. Solve this system by graphing: x + y = 5 6x + y = Solve this system by graphing: 8. Solving Linear Systems by Substitution Answers on the next page 1. Solve by substitution:. Solve by substitution:. Solve by substitution: 4. Solve by substitution: x + y = 1 y = x x + y = 7 x = y + 4 x + 5y = 5 x y = 10 7x + 4y = 1 x + y = 1 5. Solve by substitution: 6. Solve by substitution: 7. Solve by substitution: x + y = 0 4x y = x + 8y = x = 8 4y x 5 + y = 8 5 x 5 + y = Solving Linear Systems by Elimination Answers on the following page x y = 1. Solve by elimination: x + y = 10. Solve by elimination: 4x y = 1 8x = + 6y. Solve by elimination: x + y = 5 x y = 4. Solve by elimination: x y = 1 x + y =

13 5 Ÿ (a) Not a function (b) D = 0, 1, } (c) R = 4, 1, 0, 1, 4}. (a) Function (b) D =,, 4, 5} (c) R = 5, 7, 9, 11}. (a) Function (b) D = R or (, ) (c) R = R or (, ) 4. (a) Function (b) D = R or (, ) (c) R = (, 4] 5. (a) Not a function (b) D = [ 4, 4] (c) R = [ 4, 4] 6. (a) Function (b) D = R or (, ) (c) R = [0, ) 7. (a) Function (b) D =(, 4) (4, ) (c) R = (, ) p x 4x x Ÿ Yes. No. (4, ) 4. (0, ) 5. Inconsistent (no solution) 6. Dependent (innite solutions) Ÿ8. 1. (, 9). (7, ). (0, 5) 4. (, ) 5. ( ) 1 4, 1 6. Inconsistent 7. (, 1) Ÿ8. 1. (4, 6). Inconsistent 5. (0, ) 7. ( 5 ),. (1, ) 4. (, 6) 6. Dependent

14 6 5. Solve by elimination: 6. Solve by elimination: 5x + 4y = 1 x + 5y = 15 x + y = 4 x + 6y = 8 7. Solve by elimination: 6x y = x + 4y = Radical Expressions Answers on page 8 1. Find all square roots of 64.. Find all square roots of Find the square root: Find the square of: Find the square of: x Classify each root as rational, irrational, or irreal. If rational, give the exact value. If irrational, give the decimal approximation to decimal places. (a) 5 (b) 00 (c) 9 7. Find the real root: 8. Find the real root: 9. Find the real root: 10. Find the real root: 11. Simplify: 1 1. Simplify: 5 ( 9) 5 1. Simplify: x Find the decimal approximation to decmal places for this root: 4 9. Simplifying Radical Expressions Answers on page 8 Assume all variables represent positive numbers. 1. Multiply: 14 x 6. Simplify: 1. Multiply: 7x y. Simplify: 5 7. Simplify: 8. Simplify: Simplify: p Simplify: 4 81 x 4 9. Simplify: 144x y Simplify: 100m 8 z Simplify: 1x 7 y 8

15 7 1. Simplify: 4 81x 1 y Simplify: 14. Simplify: y x Rewrite radicals with the same index, then simplify: Find the unknown length of this right triangle: (, 4, c) 17. Find the distance between the points (6, 1) and (1, 1). 18. Find the distance between the points ( 8, ) and ( 4, 1). 9. Rational Exponents Answers on the next page 1. Simplify: /. Simplify: ( ) 1/ Simplify: 64 / 4. Write with radicals: 1 1 / 5. Write with radicals: (9q) 5 /8 (x) / 6. Write with radicals: (y + x) / 7. Simplify by converting to rational exponents: 1 8. Simplify by converting to rational exponents: x x 9. Simplify by converting to rational exponents: 5 t 4 t Simplify ( and ) write using positive expo- x / nents: (x ) 7 / 11. Simplify and write using positive exponents: p 1 /5 p 7 /10 p 1 / (p ) 1 /5 1. Simplify and write using positive exponents: ( ) p 1/4 q / 1 p q / 1. Simplify and write using positive exponents: k 1 /4 ( k / k 1 / ) 14. Simplify in exponential form: 15. Simplify in exponential form: x 5 x 8 4 m

16 8 Ÿ ±8 5. x ± (a) rational; 5 (b) irrational; 17.1 (c) irreal x Ÿ x 5. x 9. 1xy 4 xy 14. x x. 14xy p m 4 z 11. x y 4 1x 1. x y 4 1. y 5 y c = Ÿ or (9q) 5 (x) (y + x) 7. 1 / = 6 = x 1 / x 1 / = x 5 /6 9. t 4 / 4/5 = t 8 / p 1. 1 x 10 / q 5 / 9p 7 / 1. k 7 /4 k /4 14. x / 15. m 1 /1

17 9 9.4 Operations with Radicals Answers on the following page Assume all variables represent positive numbers. 14. Multiply & simplify: ( x + y)( x y) 1. Simplify: Simplify: Rationalize the denominator: 15. Simplify: Simplify: 8 x 8x + 7x 5. Simplify: 7x 8x 6. Simplify: Simplify: Find the perimeter of a triangle with sides of these lengths (dimensions are in inches): 0, 45, Multiply, then simplify: 5( 7 8) 10. Multiply, then simplify: ( )( + ) 11. Multiply, then simplify: ( + 1)( 1) 16. Rationalize the denominator: Rationalize the denominator and simplify: 88x 7 y9 18. Simplify: 19. Simplify: 4 y z 0. Rationalize the denominator: Rationalize the denominator: 6 5. Rationalize the denominator: x y x + y 1. Multiply, then simplify: ( 5 + ) 1. Multiply & simplify: ( + )(4 + 4). Rationalize the denominator: 1 x + y 9.5 Solving Equations with Radicals Answers on the next page 1. Solve: r =. Solve: k = 0. Solve: 9a 4 = 8a 1 4. Solve: k = k + 4k 0 5. Solve: x + 5 = 6x Solve: k + k = 1 K 7. Solve for K: V = m 1 8. Solve for L: f = π LC

18 0 Ÿ cannot be simpli- ed 4. 1 x 5. x x y x xy y yz z x xy + y x y x + y x + y Ÿ r = 11. k = 5. a = 4. k = 5 5. x = 1 6. k = 7 7. K = mv 8. L = 1 4π f C

19 1 9.7 Complex Numbers Answers on the following page 1. Write as a product of i and a simplied radical: 169. Write as a product of i and a simplied radical: 48. Simplify: Simplify: Simplify and write in a+bi form: (+i)+ ( 4 + 5i) 6. Simplify and write in a + bi form: (5 i) + ( 5 + i) 7. Simplify and write in a + bi form: ( i) + ( 4i) + (7 + 6i) 9.8 Multiplication and Division with Complex Numbers Answers on the next page 1. Simplify: (i)(7i). Simplify: ( 8i)( i). Simplify and write in a + bi form: 5i( 6 + i) 4. Simplify and write in a + bi form: (4 + 5i) 5. Simplify and write in a + bi form: (4 + i)(1 i) 6. Simplify and write in a + bi form: i( 4 i) 7. Simplify and write in a + bi form: 8. Simplify and write in a + bi form: 9. Simplify and write in a + bi form: 10. Simplify and write in a + bi form: 1 i 8i + i i + i + i i

20 Ÿ i. i i i. 4i 4. 5i i or just 0 Ÿ i i i i i i 8. + i i

21 10.1 Square Root Property, Completing the Square Answers on the following page Solve the following equations using the square root property. 1. x = 81. m =. n 7 = 0 4. (x ) = 5 5. (k + 1) = 18 Solve the following equations by completing the square. 7. x x 4 = 0 8. x + x = 0 9. w w = (x + )(x + 1) = Find the complex roots: m + 4m + 1 = 0 6. Find the complex roots: (r 5) = 10. Quadratic Formula Answers on the next page Solve these equations using the quadratic formula: x = b ± b 4ac a 1. m 8m + 15 = 0 4. Find the complex roots: x x + 6 = 0. 4k + 4k 1 = 0 5. Find the complex roots: x(x + 4) =. (r )(r + 5) =

22 4 Ÿ x = ±9. m = ±4 4. x =, 8 5. k = 1 ± 7. 4, 6} 8. x = ± x = ± i m = ± i. n = ± r = 5 ± 1 9. w = 8, Ÿ10. 1., 5}. k = 1 ±. r = 1 ± 4. x = ± i x = ± i