Inequalities With Linear Systems

Vocabulary Linear Combination Method Symbol Inequality Solution Geometry Variable Equation Algebra Greater Than Fraction Problem Solving Strategies Less Than Decimal Equal To Point of Intersection Midpoint Substitution Method Linear System Number Line

Symbols Greater Than > Less Than < Equal To = Greater Than or Equal To > Less Than or Equal To <

Number Lines

Using a Number Line What numbers satisfy this equation? 3X + 4 < 13 What numbers satisfy this equation? -What numbers satisfy this equation?

Number Line Solving 3x + 12 = 5x -4 14x -23 < 5x + 13 3w + 12 < 5w -4 3,975 + 6d < 995 + 17.95d q- 5 = 6q + 10 r – 5 > 6r + 10 3x + 17 < 47 -6x +9 < 25 18 < -4X +2 43 < 8X -9

Point of Intersection

Point of Intersection Where two lines meet/touch/cross Found using the 6 Step Method Found using the Substitution Method Found using the Combination Method

Problem Solving Strategies Make A Table Draw a Picture Solve a Simpler Problem Make an Equation Work Backwards Guess and Check

6 Step Method Set Equations Equal Get All X’s on One Side Get All Numbers on One Side X = ? Substitute X Back Into One Equation Point of Intersection

Problem 2.1 1. What kinds of equations will show how the costs for the two companies are a function of the number of days? 2. What pattern do you expect to see in graphs of the equations? 3. How can you use a graph to answer the questions about which company offers the best price?

2.1 Continued For what number of days will the costs for the two companies be the same? What is that cost? For what numbers of days will Super Locks cost less than Fail Safe? For what numbers of days will Super Locks cost less than $6000? What is the cost of one year of service from Fail Safe? How can Fail Safe adjust its per-day charge to make its cost for 500 days of service cheaper than Super Locks’ cost?

2.1 Continued B. For each company, write an equation for the cost c for days d of security services.

Problem 2.2 C = 3,975 + 6d (Super Locks) C= 995 + 17.95d (Fail Safe) Find Point of Intersection: Using 6 Step Substitution Method Using Combination Method

Applications Pg. 30 Sam needs to rent a car for a one-week trip in Oregon. He is considering two companies. A+ Auto Rental charges $175 plus $0.10 per mile. Zippy Auto Rental charges $220 plus $0.05 per mile. Write an equation relating the rental cost for each company to the miles driven. Graph the equations Under what circumstances is the rental cost the same for both companies? What is that cost? Under what circumstances is renting from Zippy cheaper than renting from A+? Suppose Sam rents a car from A+ and drives it 225 miles. What is his rental cost?

Applications Pg. 30 Maggie lives 1,250 meters from school. Ming lives 800 meters from school. Both girls leave for school at the same time. Maggie walks at an average speed of 70 meters per minute, while Ming walks at an average speed of 40 meters per minute. Maggie’s route takes her past Ming’s house. Write equations that show Maggie and Ming’s distances from school t minutes after they leave their homes When, if ever will Maggie catch up with Ming? How long will Maggie remain behind Ming? At what times is the distance between the two girls less than 50 meters?

Substitution Method Get both variables on separate sides Get either x or y completely by itself Substitute into the other equation Solve Substitute again to find the second variable value

Substitution Problems Page 56 A. 1-6

Combination Method Get X and Y on the Same Side Make Sum of 2 Equations = Sum of Totals Solve for X Plug X Back Into an Equation From Step One Solve for Y

Combination Problems Page 58 A. 1-3

Solve Using Any Method

Graphing

Pg. 40 Write each equation in y=mx + b form. x – y = 4 2x + y = 9

Parallel and Perpendicular Lines Pg. 46 Write an equation of a line parallel to the given line. Write an equation of a line perpendicular to the given line 42. y = -4x +2 36. y= 4x + 6 43. y = -(2/3)x -7 37. -6x + y = 3 44. y = 6x +12 38. x + y = 9 45. -2x + y = -1 39. x + 4y = -20 46. x – 4y = 20 40. Y = -(3/4)x -2 47. 2x + 3y = 8 41. 7x + y = -12

Parallel and Perpendicular Lines Pg. 48 Without graphing, decide whether the lines are parallel, perpendicular, or neither. 3x + 6y =12 and y = 10 + -(1/2)x Y = -x +5 and y = x +5 Y = 2 – 5x and y = -5x +2 Y = -3 +5x and y = -(x/5) +3 10x +5y = 20 and y = 10x + 20

Parallel and Perpendicular Lines Pg. 47 Suppose you are given the line equation ax + by = c. a. What is the slope of every line parallel to this line? b. What is the slope of every line perpendicular to this line?

Coordinates Tell whether each ordered pair is a solution of 3x – 5y = 15. Which equation is equivalent to 3x + 5y = 15? (-2, -4) 3x = 5y +15 (0, -3) x = -5y +5 (-10, 9) y = 0.6 x +3 (-5, -6) y = -0.6x + 3 (-10, -9) (-4, -5.4)

Problem Solving With Graphs Graph both equations Estimate a Point of Intersection Check Your Table to See if You Were Right Determine the Inequality Conditions Shade the Appropriate Area Label your Graph and Conclude

Problem Solving With Graphs Investigation 5 Page 69 -81