REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate 14 (-7)() 5(-3) A. -5 B. 11 C. 16 D. 3 E. 7 3. Factor x 4 + 8x completely A. x(x 3 + 8) B. x(x + )(x 4x + 4) C. x(x + ) 3 D. x(x + )(x 16x + 64) E. x(x + )(x x + 4) 4. Solve the systems of equations below 3x + y = 5 6x + 4y = 10 A. (1, 1) B. (-1, 4) C. Φ D. Infinitely many possibilities 5. Given the system of equations below, what is the value for x? x 5y = 11 4x + 3y = -17 A. B. - C. 3 D. None of the above 6. The graph of y = -x + 7 will intersect the graph of each of the following lines except? A. x + y = 1 B. -x + y = 14 C. 4x + y = 14 D. x + y = 10 E. x y = -17 7. What number must be added to x + 5x to make this expression a perfect square trinomial? 8. Simplify 5i 1 + 3i A. 5/4 B. 5 C. 5/ D. 5 E. None of the above A. -17 11i 10 10 B. -13 11i 10 10 C. 13 11i 10 10 D. 17 11i 10 10
9. Find the slope and y-intercept of x + 3y = 6 A. m = /3, b = 6 B. m = 3/, b = 6 C. m = 3/, b = D. m = -/3, b = E. none of the above 10. solve (5x 1) = 6 A. 1 ± 6 B. 1 ± 6 5 C. -1 ± 6 D. -1 ± 6 5 11. Multiply (3 + 5i)( 4i) 1. ( x + y)( x - y) 13. What factor is common to 9x 4 and 3x + x -? A. 3x B. 3x + C. x + 1 D. 3 E. There is no common factor 14. Which of the following is a factor of 4x + 8xy 3y? A. 4 B. x -8y C. x + y D. x y E. None of the above 15. Evaluate -4 4 A. -56 B. -16 C. 16 D. 56 16. What is the remainder when x 4 + 3x 3 x 1 is divided by x? E. Not a real number A. -5 B. -5 C. 15 D. 3 E. 35 17. Simplify: a a/b 1 1/b A. (ab a)(b 1) b B. ab b b 1 C. b 1 ab 1 D. a
18. If f(x) = 4 x-, find f(0) 19. Solve the equation below x + 3 x + 4 = _ -1 x x + 5 x + 5x A. Φ B. {0} C. {-4) D. {-4/3} 0. Given line l1 is parallel to the x-axis. Which equation has a slope that is less than the slope of l1? A. y x = 4 B. y + x = 1 C. y - 3x = 7 D. ½ y x = 10 E. y ½ x = -3 1. Simplify (x 3 5x + 8x ) (-x 3 + 5x 3x 4) A. 3x 3-10x + 5x 6 B. 3x 3 10x + 5x + C. x 3 10x + 11x + D. 3x 3 10x + 11x + (remember we add and subtract better up and down). Simplify A. 5a + 49b (5a 7b) B. 5a 70ab + 49b C. 5a 49b D. 5a 35ab + 49b 3. Solve for x 1/x = 1/a + 1/b A. x = a + b ab B. x = ab C. x = a + b D. x = ab a + b 4. Solve 5x 10x = 0 A. 0 only B. 0, /5, -/5 C. -5/, 0 D. /5, 0 E. None of the above 5. Factor: 9x 4xy + 49y A. (3x + 7y) B. (3x 7y) C. (3x + 7y)(3x 7y) D. prime 6. Solve for h A = ½ h(b1 + b) A. A ½ b1 + b B. A b1 b C. A b1 + b D. A b1 b
7. Factor completely over the set of real numbers: 3x 4 48 A. 3(x ) (x + ) 3 B. (x 4)(x + 4) C. 3(x 4)(x + 4) D. 3(x + )(x )(x + 4) 8. Simplify: 5 + 3 5 - A. 5 + 3 5 B. -3/ C. 11 + 5 5 D. 4 + 3 5 E. -1 + 5 9 9. Factor 3p + 3q px qx (completely!) A. 3(p + q) x(p + q) B. (p + q)(3 + x) C. (p + q)(3 x) D. (p q)(3 x) E. (3 x)(p q)(p + q) 30. Find the inverse function f -1 for f(x) = 4x + 3 31. Simplify completely (7x 3 ) - (1x 3 ) A. (15x 3 ) B. 3x x 3 C. x 3x D. x 15x E. x[ 7 - (1x)] 3. The perimeter of a rectangle is 4 inches, and its area is 3 square inches. Find the larger dimension of the rectangle. A. 8 in B. 16 in C. 1 in D. 4 in E. in 33. Which property is illustrated by: (8 + 9) + 10 = 10 + (8 + 9) 34. Simplify completely x 4 - x A. Distributive property B. Commutative property of addition C. Associative property of addition D. Closure property E. Identity property A. X B. X + C. x D. x E. x
35. Solve the equation x 8x + 14 = 0 A. { -7, -} B. {4 ± } C. {4 ± } D. {- 4 ± 30} E. None of the above 36. 3-3/5 = A. 8 B. 1/8 C. -8 D. -1/8 E. 8i 37. Simplify 3z + 3z z 3z 4 z 5z z + z - 15 A. 3z(z + 3) B. 3(z-3) z + 4 C. 3(z + 3) z 4 D. 3(z 3) z - 4 38. The graph shown below has a linear equation of: A. X = B. X = - C. Y = D. X + Y = E. Y = - 39. Which of the following is closest to (3.5) A. 4 B. 5 C. 6 D. 15 E. 16 40..36 1. is between: A. 1 and B. and 3 C. 10 and 0 D. 0 and 30 E. 00 and 300 F. 41. Solve for L S = N (A + L) A. L = S NA B. L = S A N C. L = S N A D. L = NA S N E. L = S N 1
REAL WORLD SCENARIOS: PART IV ANSWERS 1. 30. B 3. E 4. D 5. B 6. B 7. A 8. B 9. D 10. B 11. 6 i 1. x + (xy) y 13. A 14. A 15. A 16. E 17. D 18. 1/16 19. C 0. B 1. D. B 3. D 4. D 5. B 6. C 7. D 8. C 9. C 30. (x 3)/4 31. C 3. A 33. B 34. D 35. C 36. B 37. C 38. C 39. B 40. B 41. B
EXPLANATIONS OF REAL WO5RLD SCENARIOS: PART IV # EXPLANATIONS 1 To find the smallest value for y then x has to be at its largest amount. If you place 0 in for x, then 4x = 80. Replacing 4x with 80 4x + y = 110 80 + y = 110 subtract 80 from both sides y = 30 Please remember your Order of Operations [PEMDAS] 14 (-7) () 5(-3) (-)() + 15-4 + 15 = 11 3 Factor x 4 + 8x completely x(x 3 + 8) first factor out the x which is common, then note you have a sum of cubes x(x + )(x x + 4) then follow the pattern a 3 + b 3 = (a + b)(a + ab + b ) 4 Solve the systems of equations below 3x + y = 5 6x + 4y = 10 Recognize that the second equation is double the first. This means that they are the same line. Therefore, all solutions will work for this set. 5 Given the system of equations below, what is the value for x? x 5y = 11 4x + 3y = -17 You are asked to solve for x, therefore you need to multiply both the first and second equations by numbers that will cancel out the y values. In this case, the first will be multiplied by 3 and the second by 5, since 3 and 5 are prime. 3(x 5y) = 3(11) 5(4x + 3y) = 5(-17) 6x 15y = 33 0x 15y = -85 6x = -5 x = - 6 The graph of y = -x + 7 will intersect the graph of each of the following lines except? The only lines that DO NOT intersect are PARALLEL LINES. Parallel lines have the SAME slope. The slope of the given line is -, so see if any of the lines have the slope of -. [Hint: both the numbers in front of x and y will be positive to start off with ] 7 x + 5x + To find the c value for a perfect square trinomial, you take the b value divide it by, then square that value. (5/) = 5/4 8 Simplify 5i 1 + 3i Although we don t teach imaginary numbers in MCR, know that the rules for imaginary numbers is the same as the rule for exponents, there are no i"s allowed in the denominator, so you must remove it through multiplying the fraction top and bottom by the conjugate [same front and back, opposite sign]. Also know that i = -1 5i 1 3i = 11i + 15i = 11i 15 = -13 11i = -13-11i 1 + 3i 1 3i 1 9i 1 + 9 10 10 10
9 The trick to getting out of ALOT of work on this one is to remember how to find the slope and y-intercept without having to change the order of the equation. To find m: take A, change the sign, and divide it by B leaving it in fraction form To find b: take C and divide it by B, leave in fraction form if a fraction is the result x + 3y = 6 m = -/3 b = 6/3 = 10 (5x 1) = 6 you have to use the Square Root Property to solve this 5x 1 = ± 6 take the square root of both sides 5x = 1 ± 6 add 1 to both sides x = 1 ± 6 divide by 5 5 11 (3 + 5i)( 4i) You still use FOIL or LATTICE multiplication -4i 3 6-1i 5i 10i -0i 6 i 0i = 6 i + 0 = 6 i 1 ( x + y)( x - y) You still use FOIL or LATTICE multiplication x - y x x - xy y xy -y x + xy - y 13 9x 4 and 3x + x - to find this answer, factor both expressions completely and match 9x 4 = (3x )(3x+) 3x + x - = (3x )(x + 1) Common: 3x - 14 4x + 8xy 3y to find this answer, factor completely and match 4 (x + 7xy 8y ) 4 (x + 8y)(x y) 15-4 4 remember when doing this one, since the negative sign is not inside a set of ( ) then the answer will be negative so that eliminates of the answers, and this is a real number because it is not an even powered -56 16 This is a long division problem. Solve it that way and find the remainder x 3 + x - x + R -5 x x 4 + 3x 3 + 0x x - 1 x 4 - x 3 x 3 x 3 - x - x - x - x + 4x x - 1 x 4-5
17 a a/b treat the top and bottom separately at first 1 1/b ab a b ` b 1 b remember we are not allowed to divide fractions so: K-C-F ab a b b b 1 a(b 1) cancel out (b 1) b 1 a 18 If f(x) = 4 x-, find f(0) plug 0 in for x 4 0 = 4 - = 16-1 = 1/16 19 x + 3 x + 4 = _ -1 still have to multiply by the LCD to get rid of the fractions here (x + 5x) x x + 5 x + 5x once the fractions are gone then you can do the calculations easier (x + 3)(x + 5) x(x + 4) = -1 x + 8x + 15 x 4x = -1 combine like terms 4x + 15 = -1 subtract 15 from both sides 4x = -16 divide by 4 x = -4 0 Given line l1 is parallel to the x-axis. Which equation has a slope that is less than the slope of l1? The x-axis has a slope of 0, so the line with a negative slope is the answer. B is the only line given that has a +x in standard form, which changes to a -x in slope intercept form. 1 x 3 5x + 8x to remove the ( ), you must change all the signs in the second set of ( ) x 3-5x + 3x + 4 this now becomes an addition problem, since you changed the signs 3x 3 10x + 11x + remember to stack like terms so you don t miscalculate your answer C and D are only different by the coefficient of x 3 (5a 7b) good time to use the pattern for (a b) = a ab + b 5a 70ab + 49b 3 1/x = 1/a + 1/b LCD [abx] ab = bx + ax ab = x(a + b) ab/(a + b) = x 4 5x 10x = 0 5x (5x ) = 0 5x = 0 or 5x = 0 x = 0 or 5x = x = 0 or x = /5 5 9x 4xy + 49y check to see if this is a perfect square trinomial [(3)(7) = 4 yes] (3x 7) take square root of the first term and last term, place them in the () in the Appropriate place, then take the sign from in front of the 4 and put It between Put in square on the outside of the () and you have your answer 6 A = ½ h(b1 + b), solve for h A = h(b1 + b) multiply both sides by to eliminate the fraction ½ h = A since (b1 + b) is multiplied with h, you have to divide by (b1 + b) to b1 + b get h by itself 7 3x 4 48 factor out GCF 3(x 4 16) factor the a b 3(x +4)(x 4) factor second set of ( ) because it is also a b 3(x +4)(x )(x + )
8 5 + 3 remember that no radicals are allowed in the bottom, so X by the conjugate 5-5 + 3 X 5 + = 5 + 5 5 + 6 = 11 + 5 5 5-5 + 5 4 = 1 9 3p + 3q px qx factor by grouping 3(p + q) x(p + q) (p + q) (3 x) or (3 x)(p + q) 30 Inverse function for: f(x) = 4x + 3 remember f(x) and y are interchangeable y = 4x + 3 this means you switch the places of x and y x = 4y + 3 to find the new function, you must now solve for y x 3 = 4y y = x 3 4 31 (7x 3 ) - (1x 3 ) 3x 3x -x 3x x 3x 3 The perimeter of a rectangle is 4 inches, and its area is 3 square inches. Find the larger dimension of the rectangle. 4 = (L + W) 1 = L + W 3 = LW W = 3/L 1 = L + 3/L multiply all by L 1L = L + 3 L 1L + 3 = 0 factor now (L 4)(L-8) = 0 L = 4 or L = 8 8 is the larger dimension 33 (8 + 9) + 10 = 10 + (8 + 9) This is commutative property of addition because you are taking (8 + 9) and 10 and changing their placement around a + sign 34 x 4 factor this x factor out a -1 (x )(x + ) -1(x ) cancel out the x and the -1 moves to the top -1(x + ) x 35 x 8x + 14 = 0 is not factorable without the quadratic formula! a = 1, b = -8, c = 14 identify a, b and c; then write the quadratic formula down -b ± (b 4ac) a plug in your information -(-8) ± ((-8) 4(1)(14)) (1) this is one you should be given a basic calculator on, simplify 8 ± (64 56) 8 ± 8 8 ± 4 ±
36 3-3/5 =? note that the exponent is a fraction, -3/5 means you have to flip the 3 to the bottom, 1 Potentially cube it, then see if you can take the 5 th root. There is a slightly easier way to 3 3/5 Look at this problem break down 3 as you are dropping it to the denominator 1 ( 5 ) 3/5 this might be confusing to look at, but 3 breaks down to xxxx or 5, which means the Exponent of 5 and 3/5 actually reduces to 3 so now we have 1 = 1 3 8 37 Simplify factor and flip the second fraction at the same time 3z + 3z z 3z 4 z 5z z + z 15 3z(z + 1) (z 5)(z +3) z(z 5) (z + 1)(z 4) now remove (cancel) things that are both in the top and the bottom 3(z + 3) z 4 38 What is the equation of the line graphed? Remember that a flat line in the x,y coordinate plane does not contain an x, therefore the equation cannot contain an x. So, your answer is: y =. 39 Estimate (3.5) To do this one, think of your perfect squares that are near the given square root. 4 = 16 5 = 5 so the answer has to be very close to 5 40.36 1. is between: To estimate this: look at the whole numbers. x1 =, so the answer s smallest whole number is, therefore the choices leave you between and 3 41 Solve for L S = N/ (A + L) First get rid of the fraction by multiplying by S = N(A + L) next divide by N to remove the ( ) on the right S = A + L now subtract A from both sides to get L alone as requested N S A = L N