Math Intermediate Algebra

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1 Math Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and find its domain and range. 1. The graph of a function is shown below. Which one of the following choices represents the domain of the function? (Follow-up question: What is the range of the function?) (a) (,1] (b) (, ) (c) (,1) (d) [,] -. Which of the following tables describes a function? (a) f() (b) f() (c) f() (d) None of the tables represents a function. 1

2 3. Given the function f() = ++, which of the following represents f( 1)? (a) (b) + (c) (d) 0 Objective : Given the graph of a line, an equation of a line, or two points on a line, find the slope and -intercept of the line.. Find the slope and -intercept of the line described b the equation 5 = 10. (a) slope is 5/, -intercept is (0, 5) (b) slope is 5/, -intercept is (0, 5) (c) slope is 5/, -intercept is (0, 5) (d) slope is 5/, -intercept is (0, 5) 5. A line passes through the points (,0) and (,). Find the slope and -intercept of the line. (a) slope is, -intercept is (0, ) (b) slope is, -intercept is (,0) (c) slope is 1/, -intercept is (0, ) (d) slope is 1/, -intercept is (, 0). Find the slope and -intercept of the line shown below. (a) slope is, -intercept is (0, ) (b) slope is, -intercept is (1,0) (c) slope is, -intercept is (0, ) (d) slope is 1/, -intercept is (0, )

3 Objective 3: Find equations of lines in slope-intercept or point-slope form. Find equations of horizontal or vertical lines. Determine whether lines are parallel, perpendicular, or neither. 7. A line passes through the points (, ) and (,1). Find an equation of the line in slope-intercept form. (a) = 3 (b) = (c) = 3 5 (d) = +1. Find an equation of the horizontal line passing through the point ( 3, ). (a) 3 = 0 (b) = (c) 3 = (d) = 3 9. Two linear equations are given below. Which one of the following is true of the graphs of the equations? = = 10 (a) The graphs are parallel lines. (b) The graphs are perpendicular lines. (c) The graphs intersect at the point (0, 1). (d) The graphs are parabolas. Objective : Determine whether a sstem of two linear equations in two variables has no solution, eactl one solution, or infinitel man solutions. Solve sstems of two linear equations in two variables b using substitution, elimination, or graphical methods. 10. Solve the following sstem of linear equations. What is the -coordinate of our solution? = + (a) = (b) = 1 (c) There are infinitel man solutions. (d) There is no solution. 3 = 1 3

4 11. Solve the following sstem of linear equations. What is the -coordinate of our solution? +3 = 1 (a) = (b) = (c) There are infinitel man solutions. (d) There is no solution. = + 1. Solve the following sstem of linear equations. What is the -coordinate of our solution? + = (a) = 1 (b) = (c) There are infinitel man solutions. (d) There is no solution. 3+ = 7 Objective 5: Solve application problems that require setting up and solving a sstem of two linear equations in two variables. 13. A parking meter contains onl nickels and dimes worth \$.05. If there are eightnine coins in all, what is the value of the nickels alone? (a) \$57.00 (b) \$3.5 (c) \$5.55 (d) \$.5 1. A chemist has two concentrations of hdrochloric acid in stock: a 50% solution and an 0% solution. How much of each should she mi to obtain 100 milliliters of a % solution? (a) 0 milliliters of 50% solution and 0 milliliters of 0% solution (b) 0 milliliters of 50% solution and 0 milliliters of 0% solution (c) milliliters of 50% solution and 3 milliliters of 0% solution (d) 3 milliliters of 50% solution and milliliters of 0% solution

5 Objective : Find unions and intersections of intervals. Sketch the graph of an interval. Convert between interval notation and inequalit notation. 15. Find the union and the intersection of the intervals (,] and ( 3, ). (a) union: ( 3, ), intersection: (, ) (b) union: (, ), intersection: ( 3, ] (c) union: (, 3), intersection: [, ) (d) union: (, ), intersection: ( 3, ) 1. Which one of the following intervals is described b the inequalit < 17? (a) (,17] (b) (, ] (17, ) (c) [,17) (d) (, ] (17, ) 17. Sketch the graph of the interval described b the inequalit 5 < 1. (a) 5 1 (b) 5 1 (c) 5 1 (d) 5 1 Objective 7: Solve compound linear inequalities in one variable. Solve application problems that require setting up and solving linear inequalities in one variable. 1. Solve the following compound inequalit. Write our solution in interval notation. (a) (,1] (b) ( 1,] (c) [ 1,) (d) ( 11,1] 11 < 5 1 5

6 19. Solve the following compound inequalit. Write our solution in interval notation. (a) (, 1) [1, ) (b) ( 1,1] (c) (,] (d) (, ) [, ) +1 < 3 or Tess has \$30 and she wants to go to the fair. It costs \$10 for admission and \$1.50 per ride. How man rides can she afford? (a) 11 rides (b) 0 rides (c) 13 rides (d) 1 rides Objective : Solve linear absolute-value equations. 1. Solve for p: p = (a) p = 1 (b) p = 5 or p = 1 (c) p = 5 or p = 1 (d) p = 5 or p = 1. Solve for z: 3z +5 3 = (a) z = 0 3 or z = 1 3 (b) z = 39 3 or z = 39 3 (c) z = or z = 5 (d) z = 59 3 or z = 9 3

7 3. Solve for m: 3 m+ +7 = 5 (a) There is no solution. (b) All real numbers are solutions. (c) m = or m = 0/3 (d) m = 0/3 Objective 9: Solve linear absolute-value inequalities. Graph solutions on the number line, write solutions as inequalities, write solutions in interval notation, and write solutions in set-builder notation.. Solve the following inequalit: (a) > /5 (b) < or > (c) There is no solution. (d) All real numbers are solutions. 5. Solve the following inequalit. Graph our solution on a number line > (a) 10 (b) (c) 1 (d) 1. Solve the following inequalit. Write our solution in interval notation. (a) (,/3) (b) (, ) (/3, ) (c) There is no solution. (d) All real numbers are solutions. 3+ > 10 7

8 Objective 10: Solve inequalities and sstems of inequalities in two variables. Sketch the graphs of the solution sets. 7. Which one of the graphs shown below illustrates the solution set of the following sstem of inequalities? + < +3 (a) (b) (c) (d)

9 . Which one of the graphs shown below illustrates the solution set of the following sstem of inequalities? 1 < (a) (b) (c) (d)

10 Objective 11: Convert epressions involving rational eponents to radical epressions and vice versa. 9. Rewrite with rational eponents: ( 5 ) 15 (a) ( ) 3/5 (b) ( 5 ) 1/3 (c) ( ) 5 (d) ( ) Rewrite in radical form: (9 3 ) 7/ 7 (a) 9 (b) 9 3 (c) ( 9 3 ) 7 (d) ( ) Objective 1: Simplif radical epressions where the radicand is a monomial. 31. Assuming and represent positive numbers, simplif the following epression. (a) 3 (b) 1 1 (c) (d) 3 3. Simplif: 3 5a b c (a) 5a b c 3 a c (b) a b c 3 5a c (c) a b c 3 5a c (d) 5a b c 3 a c 10

11 Objective 13: Add, subtract, and multipl radical epressions. Simplif the results. [The epressions ma consist of sums or differences of radicals. Onl monomial radicands are considered.] 33. Epand and simplif the following epression. (a) (b) 1 (c) (d) 1 (+ )( 3) 3. Epand and simplif the following epression. (a) +9 (b) +3 (c) + (d) + +9 ( +3) 35. Simplif and combine like radicals (a) 3+ 3 (b) + 3 (c) 3 3 (d) Simplif and combine like radicals (a) (b) (c) 3 3 (d)

12 Objective 1: Divide radical epressions and rationalize denominators. [Onl monomial radicands are considered.] 37. Rationalize the denominator and simplif: n 1n (a) 1 9 (b) (c) n 3 n 3 (d) 3n n 3. Rationalize the denominator and simplif: (a) + (b) + (c) 1+ 1 (d) + Objective 15: Solve radical equations where the radicand is a linear epression. 39. Solve for : 3+ = (a) = 0 (b) = /3 (c) = /3 (d) There is no real solution. 0. Solve for : 3 = 10 (a) = /3 (b) = /3 (c) = 1/3 (d) There is no real solution. 1

13 Objective 1: Simplif comple numbers and write in standard form a + bi. [Students must be prepared to add, subtract, multipl, and divide comple numbers, as well as write square roots of negative numbers in comple form.] 1. Epand and simplif. Write our result in standard form. (a) 7i +i (b) 1i +i (c) 1+i (d) 7+i (+i)(3i 1). Epand and simplif. Write our result as a comple number in standard form. 9( 9+1) (a) 9+3i (b) 9+3i (c) 9+ 9 (d) Simplif and write in standard form: (a) 3/ (b) i (c) 5i (d) i +5i+ i + 1+i i Objective 17: Solve all tpes of quadratic equations, including those with comple solutions. Solve rational equations that reduce to quadratic equations.. Solve for : 3 = (a) = 1+i or = 1 i (b) = 1+i or = 1 i (c) = +i or = i (d) = +i or = i 13

14 5. Solve for : +5 3 = 0 (a) = or = 1/ (b) = 3 or = (c) = 1 or = 1/ (d) = 3 or = 1/. Solve for, and then find the sum of the two solutions. (a) (b) 3 (c) + 3 (d) 0 = Objective 1: Solve application problems that require setting up and solving quadratic equations. [Students should be prepared to use simple geometric formulas such as those giving the areas of rectangles and triangles.] 7. The product of two consecutive numbers is 5. If is the smaller of the two numbers, then which quadratic equation represents the problem situation? (a) +1 = 5 (b) (+1) = 5 (c) +(+1) = 5 (d) +( 1) = 5. The length of a room is ft less than twice its width. The area of the room is 0ft. Find the length of the room. (a) The length is ft. (b) The length is 10ft. (c) The length is ft. (d) The length is 15ft. 9. A garden is twice as long as it is wide. Its perimeter is ft. Find the width of the garden. (a) The width is ft. (b) The width is 1ft. (c) The width is ft. (d) The width is 11ft. 1

15 Objective 19: Graph quadratic functions. Find vertices and - and -intercepts of parabolas. 50. The function f is written below in two different, but equivalent, forms: f() = ( ) +3 f() = +11 Using either one of the forms for f, sketch the graph of f. (a) (b) (c) (d) Which of the following is an equation for a parabola that opens up and has verte (, 5)? (Each equation is written in two different, but equivalent, forms.) (a) = ( ) +5 or = +1 (b) = (+) +5 or = ++1 (c) = (+) 5 or = 1 (d) = ( ) 5 or =

16 5. Which quadratic function shown below has the given graph? (Each function is written in two different, but equivalent, forms.) (a) f() = ( 3) + or f() = +1 1 (b) f() = (+3) + or f() = 7 (c) f() = ( 3) or f() = +7 (d) None of the above Objective 0: Solve application problems involving the Pthagorean theorem. 53. The triangle shown below has the given dimensions, some of which depend on the unknown quantit. Use the Pthagorean theorem to set up and solve an equation for. 17 (a) =.5155 (b) = 1 (c) = 7 (d) = The length of one leg of a right triangle is two inches more than twice the length of the other leg. Find the perimeter of the triangle if the hpotenuse measures 13 inches. (a) 33in (b) 3in (c) 31in (d) 30in 1

17 Objective 1: Solve application problems involving direct and inverse variation. 55. When a car travels at a constant speed, the distance it travels varies directl as the time of travel. If a car travels 7 miles in hours, then how man miles does it travel in 7 hours? (a) 09mi (b) 30.5mi (c) 07.3mi (d).mi 5. The quantities and var inversel so that = 10 when =. Find when = 0. (a) = 50 (b) = 100 (c) = 5 (d) = Objective : Use substitution to solve polnomial equations that reduce to quadratic equations. 57. Solve for, and then find the sum of the two solutions. (a) (b) 1 (c) (d) 3/1 (+1) 1(+1)+15 = 0 5. In order to solve for, which is the most appropriate substitution? (a) let u = (b) let u = 3 (c) let u = 3 (d) let u = = 17

18 Free Response Problems: On our final eam ou must show all work to receive full credit for the free response problems. 59. A medieval alchemist s love potion calls for a number of ees of newt and toes of frog, the total being 1, but with twice as man newt ees as frog toes. How man of each are required? Set up a sstem of linear equations that corresponds to the problem situation and use an method to solve the sstem. 0. Graph the parabola given b the equation = ++3. Find the coordinates of the verte and the -intercepts. 1

19 1. Simplif: Solve and write our solution in interval notation. <

20 Answer Ke 1. b, Follow-up: Range is (,1] 1. b 1. c. a. d. a 3. c 3. a 3. b. a. d. b 5. a 5. a 5. d. c. b. a 7. c 7. a 7. b. b. b. b 9. b 9. d 9. d 10. d 30. c 50. d 11. c 31. a 51. b 1. b 3. c 5. a 13. d 33. c 53. b 1. a 3. d 5. d 15. b 35. b 55. b 1. c 3. a 5. d 17. a 37. b 57. b 1. a 3. b 5. d 19. d 39. d 0. c 0. a For problems 59, see our instructor for help and evaluation of our work ees of newt and 7 toes of frog 0. verte: (, 1), -intercepts: ( 3, 0), ( 1, 0) [ ), 1 Prairie State College, Department of Mathematics April 1,

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