The Keystones of Algebra 1 The Top Ten Keystones of Algebra 1 You should be able to 1) Simplify a radical expression. 2) Solve an equation. 3) Solve and graph an inequality on a number line. 4) Manipulate expressions with exponents. 5) Add, Subtract, Multiply, and Divide Polynomials. 6) FACTOR. 7) Graph a line. 8) Write the equation of a line. 9) Solve a system of linear equations or inequalities. 10) Interpret data and use basic probability.
Keystone #1 Simplify a radical expression. 68 7 144 9 54 3 72 5 132 6 85
Keystone #1 Simplify a radical expression. 1) What is the simplified form of 48? A. 3 4 B. 9 6 C. 4 3 D. 5 3 2) The diagonal of a TV screen is 275 inches long. Which expression has the same value as 275? A. 5 11 B. 3 5 C. 11 5 D. 7 11 3) The diagram below shows a square. Each side of the square measures 4 5 cm and the area of the square is 40x square cm. What is the value of x? A. 1/2 B. 2 C. 10 D. 20 5) For which value of x should the expression be further simplified? 21x A. x = 2 B. x = 5 C. x = 6 D. x = 10 A = 40x 4 5 4) A rope is tied from one corner of a fence to the opposite corner. The distance is 98 yards. Which expression shows the simplified form of 98? A. 5 2 B. 7 2 C. 8 2 D. 9 2 6) An expression is given below. 3 10a If this expression is equivalent to 90, what must be the value of a? A. 3 B. 45 C. 90 D. 115
Keystone #2 Solve an equation. 3x + 5 = 29 5(x + 7) 3 = 42 5x 3 = 7x + 5 x + 2 = 9 ½x + 7 = 3x - 18 3 x + 7-5 = 22
Keystone #2 Solve an Equation. 1) What is the solution for this equation? 2x 3 = 5 2) Which equation is equivalent to 5x 2 7x + 1 = 14x? A. x = 4 or x = 4 B. x = 4 or x = 3 C. x = 1 or x = 4 D. x = 1 or x = 3 3) The total cost (c) in dollars of renting a sailboat for n days is given by the equation c = 120 + 60n If the total cost was $360, for how many days was the sailboat rented? A. 2 C. 6 B. 4 D. 8 5) The cost to rent a construction crane is $750 per day plus $250 per hour for use. What is the maximum number of hours the crane can be used each day if the rental cost is not to exceed $2500 per day? A. 2.5 B. 7.0 C. 3.7 D. 13.0 A. 9x 2 = 14x B. 9x + 1 = 14x C. 9x + 2 = 14x D. 12x 1 = 14x 4) Solve: 3 x + 5 = 2x + 35 Step 1: 3x + 15 = 2x + 35 Step 2: 5x + 15 = 35 Step 3: 5x = 20 Step 4: x = 4 Which is the first incorrect step in the solution shown above? A. Step 1 C. Step 3 B. Step 2 D. Step 4 6) The lengths of the sides of a triangle are y, y + 1, and 7 centimeters. If the perimeter is 56 centimeters, what is the value of y? A. 24 C. 31 A. 25 D. 25
Keystone #3 Solve and graph an inequality on a number line. -2x + 5 11 5 < 2x 3 < 21 3x + 9 < 15 or 4x 2 > 18 x + 2 > 5 4 x - 3 + 1 5 ½ x + 5-8 > 1
Keystone #3 Solve and graph an inequality on a number line. 1) Which value of x is in the solution set of the inequality 4x + 2 > 10? 2) Which is a graph of the solution of the inequality 2x 1 5? A. 2 C. 3 A. 2 D. 4 3) A baseball team had $1,000 to spend on supplies. The team spent $185 on a new bat. New baseballs cost $4 each. The inequality 185 + 4b 1,000 can be used to determine the number of new baseballs (b) that the team can purchase. Which statement about the number of new baseballs that can be purchased is true? A. The team can purchase 204 new baseballs. B. The minimum number of new baseballs that can be purchased is 185. C. The maximum number of new baseballs that can be purchased is 185. D. The team can purchase 185 new baseballs, but this number is neither the maximum nor the minimum. 5. Which graph shows the solution to the inequality shown below? 15 7n 2 n 10 < 35 A. B. C. D. 4) Matthew can spend up to $60 for a carwash and gasoline at a service station. The carwash will cost $8 and the gasoline costs $3.50 per gallon. The inequality below can be solved for g, the number of gallons of gasoline Matthew can buy. 3.5g + 8 60 Which of the following is a true statement? A. Matthew can buy over 20 gallons of gasoline. B. Matthew can buy at most 14 gallons of gasoline. C. Matthew can buy 15 gallons of gasoline, but not 16. D. Matthew can buy 14 gallons of gasoline, but not 15. 6. Which graph shows the solution to the absolute value inequality shown below? A. B. C. D. 2x 7 + 3 < 12
Keystone #4 Manipulate expressions with exponents. x 3 2y 0 2y 0 x 3 y 4 2 3x 2 3 12x 2 y 4 z 3xy 2 z 3 2x 1 y 4 6xy 2
Keystone #4 Manipulate expressions with exponents. 4. 5x 3 10x 7 A. 2x 4 B. 1 5x 4 C. 1 2x 4 D. x 4 5 6. Which expression has the same value as 2 2 2 5 A. 2 10 B. 2 5 C. 2 3 D. 2 4
Keystone #5 Add, Subtract, Multiply, and Divide Polynomials. (7x 2-5x - 4) + (x 2 + 10x + 1) (2x 2 - x + 3) (7x 2 + 2x - 1) 4x(x 2-3x + 7) (x 3) 2 (2x + 5)(4x - 1) 2x + 14 x 2 + 6x 7
Keystone #5 Add, Subtract, Multiply, and Divide Polynomials. 5. Which polynomial expression represents the area of the figure shown? 6. A polynomial expression is shown below. The expression is simplified to What is the value of m? A. 2x 2 + x 10 B. x 2 + 1 x 5 2 C. 2x 2 10 D. x 2 5 A. -2 B. 2 C. 3 D. -3
Keystone #6 FACTOR. 4ab 2 + 6a 2 b 2 12a 2 b x 2 + 5x - 14 4x 2 12x 72 x 2-36 8x 2 + 18x + 9 6x 2 x - 2
Keystone #6 FACTOR. 2. Which expression represents 2x 2 12x x 6 A. 0 B. 2x C. 4x D. 2x + 2 in simplest form? 4. 5. Find the greatest common factor (GCF) for the two polynomials shown. 300ab 2 c 500a 2 bc 3 6. Find the least common multiple (LCM) for the two polynomials. A) 27xy B) 108xyz C) 108xy 2 z D) 2916xyz
Keystone #7 Graph a line. y = 2 x = -3 y = 2x - 4 y = -¼x + 4 2x + 3y = 6 x 2y = -6
Keystone #7 Graph a line. 1. What is the y-intercept of the graph 6x 3y = 24? A. 8 B. 2 2. Which set of slopes would belong to a pair of lines perpendicular to one another? A. C. 3 D. 24 B. C. D. 6. Identify whether the relation between the domain (time) and range (height) IS or IS NOT a function. I. II. A. I is a function, II is not a function B. I is not a function, II is a function C. Both I and II are functions D. Neither I or II are functions
Keystone #8 Write the equation of a line. The slope of the line is -2 and the y-intercept is (0, 5). The slope of the line is ¼ and it passes through the point (-4, -3). Slope Intercept Form: Standard Form: The line passes through the points (2, 3) and (5, 0). Slope Intercept Form: Standard Form: The line passes through the points (1, 4) and (5, -1). Slope Intercept Form: Standard Form: The line is parallel to y = 3x + 19 and passes through the point (1, -1). Slope Intercept Form: Standard Form: The line is perpendicular to the line y = 2 x 5 and 3 passes through the point (2, -2). Slope Intercept Form: Standard Form: Slope Intercept Form: Standard Form:
A. B. C. D. Keystone #8 Write the equation of a line.
Keystone #8 Write the equation of a line.
Keystone #9 Solve a system of linear equations or inequalities. x 3y = 25-5x + y = -3 x 2y = 21 3x 8y = 24 x + 2y = 6 3x 2y = 2 5x + 4y = -18 2x + 3y = -24 y + 2x = -7 2y 4 = 2x y 2x + 1 y > -3x + 4
Keystone #9 Solve a system of linear equations or inequalities. 1. Karen makes $5 per hour baby-sitting and $12 per hour giving music lessons. One weekend, she worked a total of 18 hours and made $139. How many hours did she spend baby-sitting? a) 7 hours b) 11 hours c) 18 hours d) 5 hours 2. Molly bought 3 slices of cheese pizza and 4 garlic knots for $7.40. Grace bought 2 slices of cheese pizza and 5 garlic knots for $5.75. How much does it cost to buy 2 garlic knots? a) $2.00 b) $0.35 c) $0.70 d) $4.00 3. Which of the following best describes the graph of this system of equations? y = 2x + 3 5y = 10x + 15 a) Two identical lines b) Two parallel lines c) Two lines intersecting in only one point d) Two lines intersecting in only two points 4. Sue and Maria purchased flowers. Sue purchased 5 roses and 4 daisies for $32. Maria purchased 1 rose and 6 daisies for $22. Which statement is true? a) A rose costs $1 more than a daisy. b) Sue spent $4 on each daisy. c) Sue spent more on daisies than she did on roses. d) Maria spent more on roses than she did on daisies. 5. Mark is raising money for a charity. He makes either 18 phone calls per day or 10 home visits per day. Mark wants to contact at least 100 people next week (7days). Which system of inequalities represents this situation if x represents number of days making phone calls and y is days of home visits? a) x + y 7 18x + 20y 100 b) x + y 7 18x + 20y 100 c) x + y 7 18x + 20y 100 d) x + y 7 18x + 20y 100 6. Which system of linear inequalities has the solution set shown in the graph? a) y 1 y > 3 4 x + 3 b) x 1 y 3 4 x + 3 c) y 1 y < 3 4 x + 3 d) y 1 y < 3 4 x + 3
Keystone #10 Probability, Statistics and Data Analysis. 1.) The scatterplot below compares the math and science scores of 12 students in one classroom. Based on the trend show in the line of best fit, which is closest to the expected grade of a student that scores an 86 in math? A) 82 B) 85 C) 87 D) 89 2.) Given the followings set of numbers, find the mean, median, mode and range. Student Grad e (%) 1 71 2 68 3 93 4 88 5 82 6 90 3.) There are 6 volleyballs, 2 basketballs and 5 footballs in a sports bin. If 2 balls are chosen at random, what is the probability that a football and then a volleyball is chosen if the football is replaced? 4.) A bag contains 6 red and 4 blue marbles. What is the probability of drawing a red followed by a red, if the first marble is NOT replaced? 1) 5.) Given the following box and whisker plot, determine the interquartile range. 6.) The price of the 5 most popular TV s are listed below: $400, $489, $799, $ 1,249, and $1,489. If the range was increased by $300, what could the price of the next most popular TV be?
Keystone #10 Probability, Statistics and Data Analysis. 2.) The scatter plot below compares the math and science scores of twelve students in one classroom. Based on the trend shown in the line of best fit, which is closest to the expected grade of a student that scores an 86 in math? A. 82 B. 85 C. 87 D. 89
Keystone #10 Probability, Statistics and Data Analysis. 5. The daily high temperatures in degrees Fahrenheit in Allentown, PA, for a period of 10 days are shown below. 76 80 89 96 98 100 98 91 89 82 Which statement correctly describes the data? A. The median value is 98. B. The interquartile range is 16. C. The lower quartile value is 76. D. The upper quartile value is 96. 7.) A bag contains 5 red, 2 green and 3 blue marbles. What is the probability that the next two choices will be blue if the first marble is NOT replaced? 8.) What is the Interquartile range of the following box and whisker plot? A. 9 100 B. 1 15 C. 1 10 A. 20 B. 16 C. 6 D. 7 D. 3 10
Keystone #11 Pythagorean Theorem 1. A right triangular shaped garden is planned for the front of the school. If the 2 shorter sides are 7 ft and 24 ft, what is the length of the third side? 2. The length of the hypotenuse of a right triangle is 26 inches and the length of one of its legs is 10 inches. What is the length of the other leg? 3. Which equation can be used to find the length of the rope, x, holding the tent? a) x = 8 + 6 b) x = 8 2 + 6 2 c) x = 8 2 + 6 2 d) x = 8 2 6 2 x 4. A park is in the shape of a rectangle 20 miles long and 15 miles wide. How much shorter is your walk if you walk diagonally across the park than along two sides of it? a) 25 miles b) 50 miles c) 10 miles d) 35 miles 5. A right triangle has a hypotenuse of 50 inches and a leg of 35 inches. What is the length of the other leg? a) 5 149 b) 5 51 c) 15 d) 85 6. The diagonal of a square is 20 ft. What is the perimeter of the square? a) 80 b) 40 2 c) 10 2 d) 5