2. Which of the following expressions represents the product of four less than three times x and two more than x?


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1 Algebra Topics COMPASS Review You will be allowed to use a calculator on the COMPASS test. Acceptable calculators are: basic calculators, scientific calculators, and graphing calculators up through the level of the TI86. Work out these problems and select the correct answer. 9. What are the solutions to the equation 4? a. b. ±4. c. 8 d. e. 6. Which of the following epressions represents the product of four less than three times and two more than? a. 8 b. 8 c. 8 d. 8 e. 8. If +, 4, and + represent test scores, which of the following represents the average score? a. + 4 b. 8 c d. 9 e What is the equation of the line that contains the points (,) and (6,)? a. y 6 b. y c. y d. y 7 e. y For all 6, which of the following is equivalent to a. 7 6 b. 6 6 c. 4? d. 7 6 e A board 79 inches long is cut into pieces. The second piece is inches longer than the first. The third piece is twice as long as the second. If represents the length of the first piece, then which equation [below] can be used to determine the length of the first piece? a. 79 = + b. 79 = + (+) + c. 79 = + d. 79 = + (+) + (+) e. 79 = Given the definition: a * b = a b, what is * 6? a. b. 6 c. 8 d. 6 e. 8. Write an equation to solve: The sum of consecutive odd integers is. a. ()+(+)+(+) = d. ()+(+)+(+4) = b. ()+(+)+(+) = e. ()+()+() = c. ()+()+() = 9. The statement any integer evenly divisible by two is even is: a. Always true c. True for negative integers only b. Never true d. True only if the integer is a perfect square. c. True for positive integers only. If it takes Jane hours to travel 8 miles, how long should she plan for a 4 mile trip? a. 6 hours b. 6 hours c. 7 hours d. 8 hours e. 7 hours
2 . If = , what is the value of 4? a.  b.  6 c. d. e. 4. Simplify: ( 4) (7 ) a. 4 6 d. b. 4 6 e. 6 c.. Which of the following is a factor of 8? a. + b. + 4 c. d. e Which of these is the product of ( y) and ( + y)? y a. y b. c. y y d. y y y e. y. A % fungicide solution is mied with a % fungicide solution, to get quarts of an 8% solution. How much of the % solution and how much of the % solution were used? a. 4 quarts %, 6 quarts % d. quarts %, 9 quarts % b. quarts %, quarts % e. 6 quarts %, 4 quarts % c. quarts %, quarts % ( y) 6. If =  and y =, what is the value of? y a.  b. 4 c. 8 d.  e Which of the following equations is graphed above? a. y b. y c. y d. y e. y 8. If, then = a.  b. 4 c. 4 d. e.  4 /
3 9. A car salesman received a weekly salary of W dollars plus a 6% commission on his total sales S. Which epression best describes his weekly pay? a. W + S b. (W + S) (.6) c..6w + S e. W +.6S f. W + 6S. Which of these epressions is the product of ( y )(y y )? a. y 8y 7y b. y y c. y y d. y y 8 e. y y y. Solve for : 7 a. b. c. d. e On the graph shown above, what is the distance from the point (, ) to the point (4,)? a. 4 b. 6 c. 6 d. 4 e.. John can complete a job alone in hours, and it takes Tim hours to do the same job alone. How long will it take them if they work together? a. hour b. hours c. hours d. hours e. 6 hour 4. Rationalize the denominator and simplify : a. b. 4 c. d. e. 4. The graph of the equation y =  + will lie in which quadrants of the coordinate plane? a. I and II b. I and IV c. I, II, and III d. I, II, and IV e. I, II, III, and IV 6. Subtract: 6 a. ( )( )( ) 9 b. c. 4 ( )( )( ) d. ( 6)( 9) e. /
4 Multiply: ( )( ) a. b. (+) ( ) ( ) c. ( ) ( )( ) d.  e. 8. Solve this equation: ( + ) ( ) = 4( + ) + a. =  e. = b. =  c. = d. = 9. Multiply and write your answer in a + bi form: i ( i) a. 6i b. + 9i c. 6 i d. 6 + i e.  + i. Which of the following lines is perpendicular to y 4? a. y b. y c. y 4 d. y e. y 4. How many ounces of a % alcohol solution must be mied with ounces of a % solution, to make a 4% solution? a. 4 ounces b. ounces c. ounces d. ounces e. ounces. Multiply and write your answer in a + bi form: ( + i) (4 i) a. 8 i b. + i c. + i d. 8 i e. 8 + i. Solve for : = a. = 4, =  b. =  4, = c. =  6, = d. = 6, =  e. =, = 4. Solve for : 9 9 a. = , =  b. = , =  c. = , = d. = , =  e. =, =. What is the sum of the solutions to 7? a.  b. c. 8 d. 7 e The graph of y = has an intercept of: a. (, ) b. (, ) c. (, ) d. (, ) e. (, ) h 7. The formula for the area of a trapezoid is A ( b b ), where h is the height and b and b are the lengths of the bases. If the area is 7 inches, and the bases are 4 inches and inches, what is the height? a. inches b. 6 inches c. 44 inches d. inches e. 6 inches 8. Multiply: ( 8)( 8) a.  b.  + c. d e Solve for : y z a. yz b. z y c. y z d. y z yz e. yz y z 4 /
5 4. Rationalize the denominator: 4 a. b. 9 9 c. 9 d. 9 e. 4. Multiply: a b a b 7 a. a ab b. a b c. a b a d. a b a e. a 6 ab Simplify: 7a b c a. ab c a b. 9 a b c c. abc d. ab 9 c e. a b c c 4. For a large body of water, the amount of evaporation can be modeled by: w E a. E is measured in inches per day, w is the average wind speed of the air over the water, in miles per hour, and a is the surface area of the water, in square miles. If evaporation is inch per day, and the average wind speed of the air over the water is miles per 4 hour, what is the surface area of the water? a. 9. square miles d. 9 square miles b. 4 square miles e..9 square miles c. 44 square miles 44. Simplify: a. 7 a b (a b a b ) b. 7 a b b c. a d. 7 a b e. 7 a b 4. a. < b. =, = c. <, < d. < > e. < < 46. Simplify: 8 a. 4 b..4 c. d. 6 e Find the length of the longest side (hypotenuse) of a right triangle with legs equal to 4 and. a. 9 b. 4 c. d. 4 e. y 48. Solve the system of equations: y a. =, y = b. =, y = c. =, y = 9 d. =, y = 9 e. =, y = /
6 Which of the following equations is graphed above? a. + y = 6 b. + y = 6 c. y = 6 d. y = 6 e. y = 6. Simplify: 4 a. ( ) ( 4) b. ( ) ( 4) c. ( ) d. ( ) ( ) e. ( ) ( ) (4. )(. ). Simplify: 7. a. 9 b.. c. 8. d.. e. 4.. Ten times the reciprocal of a number is less than the original number. Which equation can be used to solve for the original number? a. b. c. d. e.. What is the slope of the line + y = 6? a. b. c. d. e. 4. The formula for the surface area of a cylinder is A r r h, where r is the radius and h is the height. Which of the following is a formula for the height in terms of the area and the radius? a. A r r b. A r A A c. d. A r r e. r r r r 6 /
7 Answers to the Algebra Topics COMPASS Review. d. d. a 4. b. e 6. d 7. b 8. d 9. a. c. d. c. c 4. c. e 6. e 7. a 8. a 9. d. a. a. c. b 4. c. d 6. c 7. a 8. c 9. d. a. a. b. a 4. d. e 6. a 7. b 8. d 9. e 4. a 4. d 4. e 4. d 44. b 4. e 46. a 47. d 48. b 49. d. e. c. b. a 4. e Solutions to the Algebra Topics COMPASS Review. What are the solutions to the equation Which of the following epressions represents the product of four less than three times and two more than? four less than times is written as 4. two more than is written as +. product means to multiply. The result of all this is ( 4) ( + ). ( 4) ( + ) If +, 4, and + represent test scores, which of the following represents the average score? ( ) (4 ) ( ) 9 ( 4) 4 7 /
8 4. What is the equation of the line that contains the points (,) and (6,)? y y 7 Slope = = = 6 Method Substitute the slope and one of the ordered pairs into the pointslope formula, y y m ). 7 y = ( ) 7 y = () 7 7 y = y = y = + Method 7 Use the slope of in y = m + b. Find b, by substituting for the and for the y. 7 = () + b = 7 + b = b Now substitute both the m value and the b value into y = m + b. 7 y = + ( 4. For all 6, = 6 Factor both the numerator and the denominator, and then cancel any common factors: 4 ( 7) ( 6) 7 6 ( 6) ( 6) 6 6. A board 79 inches long is cut into pieces. The second piece is inches longer than the first. The third piece is twice as long as the second. If represents the length of the first piece, then which equation [below] can be used to determine the length of the first piece? Let = length of the first piece. Let + = length of the second piece. Let ( + ) = length of the third piece. Since all three pieces add up to 79: () + ( + ) + ( + ) = Given the definition: a * b = a b, what is * 6? a * b = a b * 6 = ( ) 6 = (4) 6 = 6 8 /
9 8. Write an equation to solve: The sum of consecutive odd integers is. An eample of consecutive integers would be,, and 7. Another eample would be,, and. Note that the second number in the set is more than the first, and the third number in the set is 4 more than the first. The algebraic interpretation is: Let = the first number. Let + = the second number. Let + 4 = the third number. sum means to add. () + ( + ) + ( + 4) = 9. The statement any integer evenly divisible by two is even is: The statement represents the definition of an even integer, and therefore it is always true.. If it takes Jane hours to travel 8 miles, how long should she plan for a 4 mile trip? 8 4 () (4) = (8) () ()(4) = 8 7 = Jane should epect her 4 mile trip to take 7 hours.. If = , what is the value of 4? 4 = ( ) 4( ) = (4) 8 = 8 =. Simplify: ( 4) (7 ) ( 4) (7 ) = 47 =. Which of the following is a factor of 8? Method Start with and at the beginning of the parentheses: ( )( ). Try different pairs of factors of 8 for the nd number in the parentheses. Then decide which will FOIL out correctly. The pairs of factors are:  and 8 and 8  and 4 and 4 Of those choices,  and 4 give the correct factorization: ( )( 4). Therefore, is a factor. Method Multiply the a and the c: times 8 = 4. Listing the pairs of factors of 4 results in:  and 4, and 4,  and, and ,  and 8, and 8, 4 and 6, 4 and 6 Choose the pair that adds to +:  and. Rewrite the quadratic epression like this: 8 8 Do factoring by grouping. ( ) 4 Factor out ( ) to yield: ( )( + 4). Therefore, is a factor. 9 /
10 4. What is the product of ( y) and ( + y)? product means to multiply. ( y)( + y) + y y y y y. A % fungicide solution is mied with a % fungicide solution, to get quarts of an 8% solution. How much of the % solution and how much of the % solution were used? Let = the amount of % solution. Let = the amount of % solution. (.) + (.) ( ) = (.8) (). +. =.8 . = = 6 There are 6 quarts of the % solution, which leaves 4 quarts as the amount of the % solution. 6. If =  and y =, what is the value of ( y)? y ( ) ( ) Which equation is graphed above? Method Note that the yintercept is (,). Move down and right to another ordered pair, such as (, 4), to discover that the slope is into y = m + b, to get y = +. Method Choose any two ordered pairs, such as (,) and (, 4). y y 4 First determine the slope with the = =.. Substitute these values /
11 Use one ordered pair and the slope of y ( ) y 8. If, = = = = 4  = with the pointslope formula, y y m ). ( 9. A car salesman received a weekly salary of W dollars plus a 6% commission on his total sales S. Which epression best describes his weekly pay? The salesman will receive W dollars, plus whatever he makes on commission. His commission is (.6)(S). Adding these together results in: W +.6S.. Epand: ( y )(y y )? ( y )(y y ) Multiply y by each of the terms in y y, and then multiply  by each of the terms in y y. y y y 8y y y 7y y. Solve for : Note that dividing by a negative value reverses the inequality sign.. What is the distance from the point (, ) to the point (4,)? Use the Distance Formula, d ( 4 ( )) ( ) = ) ( y ) d ( y with the two ordered pairs. 9 ( ) = 8 = 6 /
12 . John can complete a job alone in hours, and it takes Tim hours to do the same job alone. How long will it take them if they work together? Since John can complete the job in hours, he would get of the job done in hour. Since Tim can complete the job in hours, he would get of the job done in hour. Let stand for the time it takes them to complete the job together. Then of the job could get finished in hour. Multiply by the least common denominator 6 on both sides of the equation = 6 = 6 = 6 hours to complete the job together. 4. Rationalize the denominator and simplify : The graph of the equation y =  + will lie in which quadrants of the coordinate plane? y =  + has a yintercept of, and a slope of, so it is graphed as indicated below. Remember that Quadrant I is the upper right, Quadrant II is the upper left, Quadrant III is the lower left, and Quadrant IV is the lower right /
13 6. Subtract: Note that the least common denominator is ( + ) ( ) ( ). ( )( ) ( )( ) Multiply each fraction by one in the forms shown below, to get a common denominator. ( )( ) ( )( ) ( 6) ( ) ( )( )( ) = 4 ( )( )( ) 7. Multiply: Factor all the numerators and denominators to get ( ) ( ) ( ) ( ) ( )( ) = ( ) ( )( ) ( )( ). Cancel factors as appropriate. ( )( ) ( )( ) 8. Solve this equation: ( + ) ( ) = 4( + ) + ( + ) ( ) = 4( + ) = = = = 9. Multiply and write your answer in a + bi form: i ( i) i ( i) = i 6i = i 6() = i + 6 = 6 + i. Which line is perpendicular to y 4? The slope of the given line is. A line perpendicular to the given line will have a slope that is the negative reciprocal,. The only line with a slope of is y.. How many ounces of a % alcohol solution must be mied with ounces of a % solution, to make a 4% solution? Let = the amount of % solution. Let + = the amount of the final 4% solution. (.) + (.)() = (.4)( + ). + 4 = = = 4 ounces of the % solution /
14 . Multiply and write your answer in a + bi form: ( + i) (4 i) ( + i) (4 i) = 8 i + i i = 8 + i () = + i. Solve for : = = ( 4) ( + ) = 4 =, + = = 4, = = 4. Solve for : 9 9 ( + ) ( + ) = + =, + = =, = . What is the sum of the solutions to 7? 7 ( ) ( 4) = =, = 4 The sum of the solutions, and 4, is 7. The graph of y = has an intercept of: At the intercept, the y value will be zero. () = = = The ordered pair is (, ), since the value is, and the y value is. h 6. The formula for the area of a trapezoid is A ( b b ), where h is the height and b and b are the lengths of the bases. If the area is 7 inches, and the bases are 4 inches and inches, what is the height? h A ( b b ) 7 h (4) h 7 (4) 7 h 6 = h The height of the trapezoid is 6 inches. 7. Multiply: ( 8)( 8) ( 8)( 8) = ( )( ) = 6 6 = 4 = 4 4 /
15 8. Solve for : y z y z yz yz y z yz z y z y yz ( z y) yz yz yz or z y y z 4. Rationalize the denominator: Multiply: a b a b a b = a b 7 b 6 a = a ab = a b a 4. Simplify: 4 6 7a b c a b c a a b b b c c a a b b b c c c w 4. For a large body of water, the amount of evaporation can be modeled by: E. E is measured in inches per a day, w is the average wind speed of the air over the water, in miles per hour, and a is the surface area of the water, in square miles. If evaporation is 4 inch per day, and the average wind speed of the air over the water is miles per hour, what is the surface area of the water? w E a 4 a a (4)() 6 a a a = a = 9 /
16 44. Simplify: (a b a b ) (a = b a b ) a a b b 6 = a a bb 6 = a b 7 4. < < 4 < < 6 < < 46. Simplify: 8 Answer: a 8 = 8 Remember that 8 means 8, which is. So, we have () which is Find the length of the longest side (hypotenuse) of a right triangle with legs equal to 4 and. a b c is the Pythagorean Theorem. a and b are the lengths of the legs, and c is the length of the hypotenuse, the side opposite the right angle. a b c 4 c 6 c 4 = c 4 = c y 48. Solve the system of equations: y Use elimination: Multiply the top equation by on both the left and right sides, with the intent of eliminating the y when this equation is added to the other. ( y) () 4 y y y 7 4 = Now, substitute this value into one of the two original equations, to find y. () + y = y = The solution is: =, y =. 6 /
17 49. Which equation is graphed? Method The yintercept is . Count up and to the right from that point, (, ) to another point, such as (,). The slope is. Substitute these two values into the slopeintercept form, y = m + b, to get y =. Put this equation into standard form, since the choices are all in standard form. y y = 6 + y = 6 y = 6 Method Use the two intercepts (,), and (,), to find the slope, using m = y y = =. Substitute this slope and one of the ordered pairs into the in the pointslope form, y y m ), to get y = ( ) y = Now change the form of the problem to standard form as in Method above. (. Simplify: 4 Find the Least Common Denominator of the two denominators in this comple fraction. The LCD is ( )( ). Multiply the comple fraction by this LCD on both the numerator and the denominator. ( ) ( )( ) ( )( ) ( )( ) ) ( )( ) (4. )(. ). Simplify: 7. 8 (4.)(.) ( ) ( ) Ten times the reciprocal of a number is less than the original number. Which equation can be used to solve for the original number? Let = the original number. Then, = the reciprocal of the original number. Ten times the reciprocal of a number will be written as. less than the original number will be written as. The equation becomes. 7 /
18 . What is the slope of the line + y = 6? + y = 6 Put the equation into y = m + b form, to find m. y = y = y = The slope is. 4. The formula for the surface area of a cylinder is A r r h, where r is the radius and h is the height. What is the formula for the height in terms of the area and the radius? A r r h A r r h A r r h / 8 /
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