LINEAR INEQUALITIES: Absolute Value NOTES: This number line illistrates the inequality: |x| < 7 All of the points on the highlighted interval are less than 7 units from 0 on the number line. This number line illustrates the inequality: |x| > 5 All of the points on the highlighted parts are more than 5 units from 0 on the number line. This number line illustrates the inequality: |x-4| < 7 All of the points on the highlighted interval are less then seven units from four. This number line illustrates the inequality: |x-7| > 8 All of the points on the highlighted parts are more than 8 units from 7. *** The inequality |x| < 7 is equivalent to the compound statement -7 < x AND x < 7. The inequality |x| > 7 is equivalent to the compound statement x < -7 OR 7 < x. *** The inequality |x-8| < 6 is equivalent to the compound statement -6 < x-8 AND x-8 < 6. The inequality |x-7| > 6 is equivalent to the compound statement x-7 < -6 OR 6 < x-7. *** Let's solve for x: x + 7 < 14 (x + 7) - 7 < 14 -7 x < 7 Let's solve for x: x - 3 < 12 (x - 3) + 3 < 12 + 3 x < 15 *** Now we'll solve another: 2x < 8 (2x)/2 < 8/2 x < 4 Let's solve this inequality: -x < 12 Add x to both sides: -x + x < 12 + x 0 < 12 + x Subtract 12 from both sides: -12 < x Here's another: -9 > -x Add x to both sides: -9 + x > -x + x -9 + x > 0 Add 9 to both sides: x > 9 *** To solve for x: 4x - 9 < 3 Add 9 to both sides: 4x < 12 Divide both sides by 4: x < 3 *** The number line illustrates the inequality: |x-7| > 8 All of the points on the highlighted parts are more than 8 units from 7. *** The inequality |x| < 7 is equivalent to the compound statement -7 < x AND x < 7. The inequality |x| > 7 is equivalent to the compound statement x < -7 OR 7 < x. *** The inequality |x-8| < 6 is equivalent to the compound statement -6 < x-8 AND x-8 < 6. The inequality |x-7| > 6 is equivalent to the compound statement x-7 < -6 OR 6 < x-7. *** Let's solve for x: x + 7 < 14 (x + 7) - 7 < 14 - 7 x < 7 Let's solve for x: x - 3 < 12 (x - 3) + 3 < 12 + 3 x < 15 *** Now we'll solve another: 2x < 8 (2x)/2 < 8/2 x < 4 *** Let's solve this inequality: -x < 12 Add x to both sides: -x + x < 12 + x 0 < 12 + x Subtract 12 from both sides: -12 < x Here's another: -9 > -x Add x to both sides: -9 + x > -x + x -9 + x > 0 Add 9 to both sides: x > 9 *** To solve for x: 4x - 9 < 3 Add 9 to both sides: 4x < 12 Divide both sides by 4: x < 3 *** The inequality |x-5| < 7 can be solved by first rewriting it into the compound statement: -7 < x -5 AND x - 5 < 7 and then finding the x values that satisfy these inequalities: -2 < x AND x < 12 *** SYSTEMS OF EQUATIONS AND INEQUALITIES: Problem Set 1 NOTES: BUS: The conditions now placed on the buses are: x <= 7 y <= 4 x + y <= 9 Additionally, x >= 0 and y >= 0, since we cannot have a negative number of buses! The points in the graph are the only possible solutions since we can't have a fraction of a bus! State laws just won't allow a school bus to operate without the front half. ;-) In fact, these 3 points represent the only possible solutions. Think about why the other points are excluded from the solution. Only the points highlighted would use 9 total buses, the maximum possible. *** FOOTBALL: It's fourth quarter in the homecoming football game. With a minute left in the game your team is down by 4 points and must score a touchdown in order to win. The coach is sure it will take at least 2 passing plays and 1 running play to score. The quarterback has time for no more than 5 plays. How many winning plays is possible? 6! Here's the graph of x >= 2 AND y >= 1 AND x + y <= 5. Since the number of football players must be a whole number we see there are 6 winning combinations: (2,1),(3,1),(4,1),(2,2),(3,2),(2,3) ANS: x<=7 , y<=4 , x+y<=9 , #4 , 6 *** SYSTEMS OF EQUATIONS AND INEQUALITIES: Problem Set 2 NOTES: POST OFFICE: The post office limits the size of envelopes it will accept at regular rates. If x represents the width and y the height (both in inches), then to be acceptable these conditions must be met: 5y >= 2x AND 9y <= 7x Will an envelope 3 inches wide and 5 inches tall be accepted? x = width y = height NO. What limitations are places on the width if the height is 3 inches? These are the correct answers: 5(3) >= 2x 9(3) <= 7x 15/2 >= x 27/7 <= x 7 1/2 >= x 3 6/7 <= x From the graph it appears that, within limits, the width and the height can increase infinitely. So the post office added these limitations: 5 <= x AND x <= 11 1/2 3 1/2 <= y AND y <= 6 1/8 Will an envelope 11 1/2 inches wide and 8 inches tall be accepted? NO. Will an envelope 6 inches wide and 4 inches tall be accepted? YES. *** THE UN: A committee being formed at the United Nations was to have at least 2 members from the Western Hemisphere and at least 4 from the Easter Hemishpere. Additionally, a hemisphere was to have no more than 5 members on the committee. How many possible ways are there to form the committee? 8. Here's the graph of x >= 2 AND y >= 4 AND x <= 5 AND y <= 5. Since the number of committee members must be a whole number, we see there are 8 possible committees: (2,4),(3,4),(4,4),(5,4),(2,5),(3,5),(4,5),(5,5) WORKSPACE: 5y >= 2x AND 9y <= 7x x = width y = height W: - H: - ANS: NO , YES , 7 1/2 | 3 6/7 , NO , NO , NO , YES , 8 *** SYSTEMS OF EQUATIONS AND INEQUALITIES: The Function Machine NOTES: FUNCTION MACHINE: Find the expressions: A finite difference game for: ax + b This game has two forms. - You can choose a secret expression that the function machine will find. - You can find a secret expression that the function machine has chosen. In this activity, you will pick an expression of the form: ax + b The function machine will ask for the value of the expression when x=0 AND x=1. Then the function machine will give you the expression. This is an example of a game. Suppose you pick the expression 4x + 5. The function machine will have you fill in values in a table. 4x + 5 | The value X | of your | expression ---+----------- 0 | 5 1 | 9 Next, the funtion machine will give the value of your expression when x=2 and give you the secret expression. Let's try it a couple of times!! On a piece of paper write an expression of the form ax + b. Please choose numbers that are between -30 and 30. Ex: 2x + 6 AND 7x - 3 *** I picked: 18x - 2 IT GOT IT RIGHT!! AMAZING!! *** You will examine the values of the expression ax + b You will use differences of values for the expression to find the secret expression. This method of finding an expression of the form ax + b is called the method of FINITE DIFFERENCE. Study the table: X | Value of | ax + b ---+--------- 0 | a(0) + b = b \ | | a + b - b = a 1 | a(1) + b = a + b / When x=0, the value is b. When x=1, the value is a+b. The difference is a. x = 0, OUTPUT = 4 x = 1, OUTPUT = 9 Complete the expression: Form: ax + b Expression: 5x + 4 When x=0, ax + b = a(0) + b = 4 - so b must be 4! Now you can figure out the rest... When you chose any two consecutive whole numbers for x, the difference is a. Think about how to find b. An easy way to find a and b in an expression of the form ax + b, is to use x=0 and x=1. WORKSPACE: ANS: 5 , 9 , 4 , 5 , -9 , -7 , a , a , a ,