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Tutorial for the Algebra 2A/B Summer Assignment ****At the end of the tutorial, there are example problems to be completed. Ex. 1 Solve 6 3x 9 27 +9 +9 Add 9 to each section. +9 3 3 x 36 3 3 3 Divide by 3. 1 x 12 Graph < or > use an open circle when you graph or use a closed circle 3 8 x 12 3x 53x 6 2 x 7 Distribute 3 to(8 x 12) and 4 4 5to(3 x 6) The result is 6x 9 3x 15x 30 2x 7 Combine Like Terms Ex. 2 Solve 9x 9 13x 37 13x 13x 22x 9 37 +9 +9 22 x 28 Solve for x. Divide by 22 and reduce if possible. 22 x 28 22 22 x 14 11 Graph Ex. 3 Solve and Graph 4x 7 9 For greater than absolute value inequalities you need to create two inequalities. First copy the original problem without the absolute value symbol. The second inequality is created by reversing the inequality and taking the opposite sign of the # on the right. Your inequalities will look as follows: or Solve the inequalities. 4x 7 9 4x 7 9 4x 7 9 +7 +7 or 4x 7 9 +7 +7 4x 16 4 x 2 4 x 16 4 4 4x 2 4 4 x4 or x 1 2 Graph This is an or problem because the inequality is either > or , which means when we graph the solution we take the union of the two graphs. Ex. 4 Solve and graph 5x 6 9 or +6 +6 3x 5 38 -5 -5 5 x 15 5 5 or 3 x 33 3 3 x3 or x 11 Graph Ex. 5 Solve and graph 3x 2 8 You also need to create two inequalities for a less than absolute value inequality, but you will write it as one single expression. To do this you must begin by writing the original inequality without the absolute value symbol, than place the opposite sign of the # to the right in front of the original inequality as shown below. 8 3x 2 8 -2 -2 -2 Follow Ex.1 to help you solve the inequality. 10 3 x 6 3 3 3 Graph Ex. 6 Solve and graph 4 x 5 13 Follow the steps from Ex. 3. 4x 5 13 -5 -5 or 4x 8 x2 4x 5 13 -5 -5 4 x 18 or x 18 reduce 4 x 9 2 Graph Ex. 7 Solve. 3x 2 17 This is an absolute value equation. You create two equations by writing the original equation without the absolute value symbol, and the second equation is the same except you take the opposite sign of the number on the right. See below. 3x 2 17 -2 -2 or 3x 2 17 -2 -2 3x 15 3x 19 3 x 15 3 3 3x 19 3 3 x x5 Ex. 8 Solve. 7 3x 28 7 3x 28 -7 -7 19 3 Follow the steps from example 7. or 7 3x 28 -7 -7 3x 21 3x 35 3 x 21 3 3 3 x 35 3 3 x7 x 35 3 For examples 9 – 11, write the equation of the line, then graph the line. Ex. 9 Given two points (5,3) and (-2,10), find the equation of a line. First find the slope given by the formula m y 2 y1 10 3 7 1 . m = 25 7 x2 x1 Next use the slope-intercept form of an equation which is y mx b where m = the slope and b = the y-intercept. Substitute in for x and y using any given point and substitute in for m from step one to find b. y mx b 3 1(5) b 3 5 b 8b Now substitute in for m and b into the equation y mx b. Answer: y x b Ex. 10 x-intercept is -4 This is the point (-4,0). This is the point (0,7). y-intercept is 7 Now find the slope and follow the steps in example 9. m 70 7 0 (4) 4 y mx b I chose to use point (0,7) 7 ( 0) b 4 b7 7 Answer: y 7 x7 4 Ex. 11 f (4) 3, y x Point 1 (4,-3) f (2) 5 x y Point 2 (2,5) Now find the slope and follow the steps in example 9. m 5 (3) 8 4 24 2 y mx b 5 4 ( 2 ) b 5 8 b 13 b Answer: y 4 x 13 Ex. 12 You want to fence in your yard. The area that you want to enclose is 800 square feet. The length of the property is 40 feet. How much will it cost you to put a fence around your property if fencing costs $35 a foot? Area = length width A lw 800 40 w 800 40 w 40 40 20 w 40 20 20 40 Now find the perimeter: P 2l 2w P 2(40) 2(20) P 80 40 120 feet The cost for fencing will be 120 feet $35 = $4,200. Ex. 13 A plumber charges $65 to enter your home and $45 per hour of work. a) Write an inequality that represents the possible number of hours the plumber could work for $335. $65 represents your fixed cost, $45x represents the number of hours worked times $45, and $335 represents the maximum cost, thus the following equation: $65 45x $335 b) Solve the inequality: 65 45x 335 -65 -65 45 x 270 45 x 270 45 45 Answer: x 6 Ex. 14 The length of a soccer field can vary from 80 feet to 120 feet inclusive. Write an inequality that describes the possible lengths of a soccer field. Answer: 80 x 120 Ex. 15 On your first five tests of the marking period, you earn grades of 95, 90, 80, 88, and 85. What grade would you need to earn on your last test so that you have an average of 88. Since you will have taken six tests and the average of the six tests is an 88, multiply 6 88. This will give you the total points needed for an 88 average after 6 tests. Total points will equal 528 (6 88=528). Now take the sum of the five tests and subtract that total from 528. 95+90+80+88+85=438 Total Points: Minus the sum of the 5 scores: 528 -438 Score needed to average an 88: 90 Answer: You need a 90 on your last test to get an 88 average. Ex. 16 Graph the function f ( x) mx b m = the slope = f ( x) is the same as rise run 2 x 5. 3 y mx b b = the y-intercept, which is the 1st point you plot on the y-axis. Step 1: Plot -5 on the y-axis. Step 2: From -5, the slope tells you to go up two and right 3 where you will plot your 2nd point. Connect 2 the two points and you now have the graph of f ( x) x 5 3 Ex.17 Graph the following inequality: y 5 x 3 m the slope = rise 5 run 1 b y int ercept 3 Follow Ex. 16 for directions on how to graph y 5 x 3. Shade below the line because it is less than or equal to. or is a solid line because the points on the line are included. < or > is a dashed line because the points on the line are not included. Ex.18 Here is an example of a vertical line test. a. b. This is an example of a graph that passes the vertical line test. When you draw vertical lines they intersect at only one point. This graph fails the vertical line test. When you draw vertical lines they intersect at more than one point. Ex. 19 Graph the following absolute value function: y 2 x 2 3 . This is in the form of rise . To graph the function, start at point (0,0). H moves the point (0,0) run left or right. For ex., x 2 moves the graph 2 units to the right. y a x h k . a = the slope = x 2 moves the point 2 units to the left. Wherever h lands on the x-axis, move up or down according to what k is. 2 moves the point up 2 units and -2 down 2 units y 2 x 2 3. From (0,0) move left 2, then down 3 and plot your first point at (-2,-3). rise a = the slope = . From the point (-2,-3) the slope tells you to rise 2 then plot 2 additional points left run 1 and right 1 to form a v figure. See the graph on the following page. Ex. 20 Write the absolute value function that has the following properties: a) Reflects over the x-axis: this means the graph opens downward. b) Has a vertical stretch of 3: this means the slope equals a = 3 . 1 c) Has a horizontal shift of 4: this means 4 the left from (0,0) and k = 4 d) Has a vertical shift up 4: from -4 on the x-axis move up 4 then plot the point (-4, 4) Now substitute in the equation y a x h k therefore y 3 x (4) 4 . Now simplify to get y 3 x 4 4 . Ex. 21 Evaluate the following function for x = -2: f ( x) 4 x 2 3 x 12 f (2) 4(2) 2 3(2) 12 f (2) 4(4) 6 12 Evaluate is another word for substituting in for x. f (2) 16 6 12 f (2) 2 Follow the order of operations (PEMDAS). Ex. 22 At 4AM, there was 3 inches of snow on the ground and at 2PM there were 23 inches of snow on the ground. a) Write a model for this situation. First, find the slope using: m = (the difference in snowfall) (the difference in time). m 23 3 20inches 2inches / hour. 2 PM 4 AM 10hours The linear model is y 2 x 3 ; 2x represents the snowfall/hour and 3 represents the original amount of snow. b) Based on your model, how much snow was there at 4PM? From 4AM to 4PM is 12 hours and x = hours, therefore: Ex. 23 Graph the line 5 x 3 y 15. Determine the x- and y- intercepts. Also, determine the slope of the function. To find the x-intercepts, set y = 0, then solve for x. 5 x 3(0) 15 5 x 15 Plot 3 on the x-axis. x3 To find the y-intercept, set x = 0, then solve for y. 5(0) 3 y 15 3 y 15 Plot -5 on the y-axis, then connect the points to get your graph of a line. y 5 Now to find the slope of the equation put the equation into slope-intercept form ( y mx b ) by solving for y. 5 x 3 y 15 The slope is m 3 y 5 x 15 y 5 x5 3 5 3 Ex. 24 Solve the following absolute value equation: y 4 x 7 12 for y 16. 16 4 x 7 12 28 4 x 7 7 x7 First substitute 16 in for y. Then add 12 to both sides. Then divide by 4. Now, follow the steps from example 7. x7 7 or x 7 7 x = 14 or x=0 Practice Problems Solve, state the solution set, and graph. SHOW ALL WORK!!!! 1. 5 10 x 15 25 4 2. (20 x 45) 3x 4(7 x 4) 8 x 2 5 3. 3x 5 10 4. 4 x 3 9 or 7 x 1 43 5. 4 x 1 17 6. 3x 2 11 Solve the following absolute value equations 7. 7 x 5 9 8. 5 4 x 16 Write the equation of a line that passes through the given points. Then, graph the line. 9. (7, 2) and (-4, 6) 11. f (3) 2, f (8) 8 10. x-intercept 5 y-intercept -6 SHOW ALL WORK!!!!! 12. You want to fence in your yard. The area that you want to enclose is 600 square feet. The length of your property is 20 ft. How much will it cost you to put a fence around your property if fencing costs $25 a foot? 13. A plumber charges $75 to enter your house and $40 per hour of work. a. Write an inequality that represents the possible number of hours the plumber could work for $515. b. Solve the inequality. 14. The length of a standard hockey rink can vary from 80 ft. to 90 ft., inclusive. Write an inequality that describes the possible lengths of a hockey rink. 15.On your first four tests of the marking period, you earn grades of 98, 93, 82, and 87. What grade would you need to earn on your last test so that you have an average of 90? 16. In your own words, define what a function is. Then, graph the function: f ( x) 3 x 3. 4 17.Graph the following inequality: y 4 x 1 . Shade the appropriate region. 18.What is the vertical line test and how is it used? Write your answer using complete sentences. Give an example of a graph that passes the vertical line test and an example of a graph that fails the vertical line test. 19.Graph the following absolute value function: y 3 x 1 4 20.Write the absolute value function that has the following properties: a. b. c. d. Reflects over the x-axis. Has a vertical stretch of 7. Has a horizontal shift of 9 to the right. Has a vertical shift down 2. 21.Evaluate the following function for x = 3 : f ( x) 2 x 6 x 9 2 22. At 6AM, there was 6 inches of snow on the ground and at 1PM, there was 28 inches of snow on the ground. a. Write a model for this situation. b. Based on your model, how much snow was there at 5 PM. 23.Graph the line 4 x 2 y 8 . Determine the x- and y-intercepts. Also, determine the slope of the function. 24. Solve the following absolute value equation: y 3 x 5 6 for y = 9.