Park Forest Math Team. Meet #3. Algebra. Self-study Packet

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1 Park Forest Math Team Meet #3 Self-study Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number Theory: Divisibility rules, factors, primes, composites 4. Arithmetic: Order of operations; mean, median, mode; rounding; statistics 5. : Simplifying and evaluating expressions; solving equations with 1 unknown including identities

2 Important information you need to know regarding ALGEBRA Absolute value; inequalities in one variable including interpreting line graphs Absolute Value is the distance a number is from zero. Absolute value is never negative. The symbol for absolute value is and Inequalities To solve an inequality, solve as if it were a regular equation. Remember to switch the direction of the inequality sign only if you multiply or divide by a negative!

3 Category 5 Meet #3 - January, ) What are the two values of N that make this absolute value equation true? 2) The graph represents the solution set of the inequality - 3 X What is the value of A + B? A B 3) Solve the following inequality for C: 5(C + 2) - 4(2C - 3) + 7(3C - 8) < - 6(0.5C - 9) - 64 Express your answer as a common fraction. ANSWERS 1) and 2) A + B = 3) C <

4 Solutions to Category 5 Meet #3 - January, ) - 5 ; 19 (any order) 2) ) 1) Either N - 7 = 12 or N - 7 = So, either N = 19 or X = ) original inequality having added 17 to both members having divided both members by - 3 So, A + B = = ) 5(C + 2) - 4(2C - 3) + 7(3C - 8) < -6(0.5C - 9) - 64 original inequality 5C C C - 56 < -3C distribute 18C - 34 < -3C - 10 combine terms 21C < 24 add 3C; add 34 C < 8/7 divide 21

5 Category 5 Meet #3 - January, th anniversary edition 1) If C = 5, then what is the sum of all values of C that make this sentence true? 2) The graph below represents the set of all values of X that make X - N 8 true. What is the value of N? X 3) If 7 + 3(2Y - 5 )< 52 and then the set of all possible values of Y is represented in the graph below: A What is the value of A + B? B ANSWERS 1) 2) 3)

6 Solutions to Category 5 Meet #3 - January, ) The two solutions are 5 and -5. Their sum is zero. 1) 0 1) 2) The absolute value sentence can be 2) 20 translated as, "The distance between a number, N, and all solutions is at most 3) 15 8 units." N can be found by locating the midpoint of 12 and 28, which is 20. 3) 7 + 3(2Y - 5 )< Y - 15 < 52 6Y - 8 < 52 6Y < 60 Y < 10 also, 7-3(2Y - 5 )< Y + 15 < Y < - 8-6Y < - 30 Y > 5 Therefore, A = 5 and B = 10 and their sum, A + B, is = 15.

7 Meet #3 January 2012 Category 5 1. How many integers do not satisfy the inequality below? 2. Find the positive difference between the two solutions to the equation: 3. The graph below describes the solution to the inequality: Find the value of

8 Meet #3 January 2012 Solutions to Category 5 1. Let s solve the inequality: If the argument is positive we get: If the argument is negative we get:, so the solution to the inequality is { } What integers do not fall in this range? { } - a total of integers. 2. In the positive case: we get or In the negative case: we get or The differnce between the two solutions is. 3. The graph depicts all the points on the number line whose distance from is no more than. In other words, it is the visualization of:, which makes. The abosolute value function measures the distance between points on the line.

9 Category 5 - Meet #3, January What is the positive difference between the least and largest integers that satisfy the inequality below? x What value of the parameter M in the inequality M x 1 3 will make the solution agree with the line graph below? Express your answer as a decimal For how many integers N (excluding zero) does the inequality below hold true? 30 N > 2 N

10 Solutions to Category 5 - Meet #3, January x can be written (multiply both sides by 5) as x If the argument on the left is positive we get x 5 10 so x 15. If the argument is negative we get x 5 10 so x 5. Taken together the range of values that make the inequality true are: 5 x 15. The largest integer solution is 15, the least is 5, and the difference is therefore If the argument in the inequality M x 1 3 is positive then we can write: M x M 3 Or x 3+M = and if the argument is negative we can write: M M M M x 3 Or x 3 M = M 3 = 1 3 M M M Comparing these to the graph we require that = 3 AND 1 3 = 1 M M The solution to both is 3 M = 2 Or M = 3 2 = 1.5. Alternatively, by looking at the graph solution we can ask ourselves: What inequality is described by the graph? And the answer is that the graph depicts points outside the region (-1, 3), or in other words points whose distance from the middle of the region is at least 2 (half the size of the region). So we can conclude that the graph depicts the inequality x 1 2 (1 being the midpoint of the region (-1, 3)). Comparing this inequality to the one given we can again calculate that M = 1.5 is the correct value. 3. Since both sides of the inequality are positive let s see what happens when N is positive: 30 N > 2 N means N2 < 15 and so N < 4 or in other words N can take on the values {1, 2, 3}. If we assume N is negative, the same will apply (no need to reverse the inequality, as both sides are positive due to the absolute value), so N can take on the values { 1, 2, 3}. Overall there are 6 possible values for N.

11 Category 5 Meet #3, January What is the positive difference between the two solutions to this equation? 6 3x = For what value of K is the solution set of the inequality below given by the graph below? For how many integer values of y is 18 2 y a positive integer?

12 Solutions to Category 5 Meet #3, January or or 18 2 or 6 The positive difference between 2 and 6 is 6 ( 2) = This inequality becomes an equality at the border point Since is an integer, 2 will be an integer. In order for to be a positive integer, 18 must be divisible by 2 but you don t need to worry about the positive part since the absolute value bars will take care of that. Basically we want to know how many values of y make 2 a factor of 18 remembering that So 2 18, 9, 6, 3, 2, 1, 1, 2, 3, 6, 9, 18 which means that 16, 7, 4, 1, 0, 1, 3, 4, 5, 8, 11, 20 for 12 integer values.

13 Category 5 Meet #3, January How many integer values of n satisfy the inequality below? 18 n + 1 > 3 2. Find the sum of the two solutions to the equation below. Express your answer as a mixed number with the fraction part in lowest terms. 3x = For what value of B does the solution to the inequality below match the graph below? 3( 5x 4) 3x + B >

14 Solutions to Category 5 Meet #3, January There are ten (10) integer values of n that satisfy the inequality. They are -6, -5, -4, -3, -2, (not 1), 0, 1, 2, 3, and For example, when n = -6, we get = 18 5 = 18 5 = 3 3 5, which is greater than First let s subtract 9 from both sides of the equation. 3x 7 = 33 The expression inside the absolute value bars could be either negative or positive, so we have to solve the two separate equations below. 3x 7 = 33 3x = 40 x = 40 3 or 3x 7 = 33 3x = 26 x = 26 3 The sum of these two solutions is = = Let s simplify the inequality first. 3( 5x 4) 3x + B > 21 15x 12 3x + B > 21 12x 12 + B > 21 12x > B 12x > 9 B x > 9 B 12 We know from the graph that x > 3. This means that 9 B 12 must equal 3, or equivalently 9 B = 36. Solving for B, we get B = 9 36 = 45. Alternatively, one could substitute 3 for x from the beginning and then solve for B.

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