Regents Review #4 Inequalities and Systems

Simple Inequalities 1)Solve inequalities like you would solve an equation (use inverse operations and properties of equality to isolate the variable). 2)When multiplying or dividing both sides of an inequality by a negative number, reverse (flip) the inequality sign. 3)Graph the solution set on a number line.

Simple Inequalities -3x – 4 > 8 -3x > x < - 4 3(3x – 1) + 3x > 5(2x + 1) 9x – 3 + 3x > 10x x – 3 > 8x + 4 4x > 7 x > (1.75)

Simple Inequalities Words to Symbols At Least Minimum Cannot Exceed At Most Maximum Example In order to go to the movies, Connie and Stan decide to put all their money together. Connie has three times as much as Stan. Together, they have more than $17. What is the least amount of money each of them can have? Let x = Stan’s money Let 3x = Connie’s money x + 3x > 17 4x > 17 x > 4.25 Since Stan has to have more than $4.25, the least amount of money he can have is $4.26. Since Connie has three times as much as Stan, she has $12.78.

Simple Inequalities Erik and Julie, an engaged couple, are trying to decide which venue to use to hold their wedding reception. Venue A charges a $2500 site fee in addition to $45 per person. Venue B charges a $3200 site fee plus $40 per person. Using an inequality statement, determine the minimum number of people who must attend the wedding in order for venue B to be more cost effective than venue A. x: # of people Venue A: x Venue B: x Venue B < Venue A x < x 40x < x -5x < -700 x > 140 At least 141 people must attend the wedding for venue B to be more cost effective. # of people (X) Venue A ( Y 1 ) Venue B ( Y 2 )

Compound Inequalities A compound inequality is a sentence with two inequality statements joined either by the word “OR” or by the word “AND” “AND” Graph the solutions that both inequalities have in common (overlap) The solutions must make both inequalities true “OR” Graph the combination of both solution sets The solutions only need to make one inequality true

Compound Inequalities “AND” Solve -12 2x < -8 2x -12 and 2x < -8 x -6 and x < x < The temperature today will be 42⁰ plus or minus 5⁰. Write and graph a compound inequality to represent all the temperatures of the day. Let x = the temperatures for the day. 42 – 5 < x < ⁰ < x < 47⁰

Compound Inequalities “OR” Solve the inequality 2x In order to participate in the big buddy/little buddy bowling league, you must be at least 18 years old or under 10 years of age. Write and graph a compound inequality to represent all the ages of people who participate in the program. Let x = ages of people in the program x 18 2x 5 x < 3 x 5

Linear Inequalities Graph Linear Inequalities in two variables the same way you graph Linear Equations in two variables but… 1)Use a dashed line (----) if the signs are 2)Use a solid line ( ) if the signs are or 3)Shade above the line if the signs are > or 4)Shade below the line if the signs are < or SEE FLIP #11 ON HALGEBRA.ORG

Linear Inequalities Graph -2y > 2x – 4 -2y > 2x – y < - x + 2 m = b = 2 (0,2) Test point (0,0) -2y > 2x – 4 -2(0) > 2(0) – 4 0 > 0 – 4 0 > - 4 True -2y > 2x - 4 Calculator Checks with Inequalities: See FLIP #11

Systems A "system" of equations is a collection of equations in the same variable When solving Linear Systems, there are three types of outcomes… No Solution y = 2x + 5 y = 2x – 4 One Solution y = -2x + 4 y = 3x - 2 Infinite Solutions y = 2x + 3 3y = 6x + 9

Systems There are two ways to solve a Linear System 1)Graphically-graph both lines and determine the common solution (point of intersection) 2)Algebraically -Substitution Method -Elimination Method

Systems y = 4x – 1 m = b = -1 (0,-1) 3x + 2y = 20 2y = -3x + 20 y = - x + 10 m = - b = 10 (0,10) Solution (2,7) 3x + 2y = 20 Y = 4x – 1 Check (2, 7) y = 4x – 1 3x + 2y = 20 7= 4(2) – 1 3(2) + 2(7) = 20 7 = 8 – = 20 7 = 7 20 = 20 Solve the system y = 4x – 1 and 3x + 2y = 20 graphically.

Systems Andy’s cab Service charges a $6 fee plus $0.50 per mile. His twin brother Randy starts a rival business where he charges $0.80 per mile, but does not charge a fee. a)Write a cost equation for each cab service in terms of the number of miles. b)Graph both cost equations. c) For what trip distances should a customer use Andy’s Cab Service? For what trip distances should a customer use Randy’s Cab Service? x = the number of miles C = the cost Andy’s C(x) = 0.5x + 6 Randy’s C(x) = 0.8x If the trip is less than 20 miles, use Randy’s cab service. If the trip is more than 20 miles, use Andy’s cab service. If the trip is exactly 20 miles, both cabs cost the same amount. [Check algebraically 0.5x + 6 = 0.8x]

Systems Solving Linear Systems Algebraically (Substitution) x + y = 7 3x = 17 + y Finding y 3x = 17 + y 3(7 – y) = 17 + y 21 – 3y = 17 + y -4y = -4 y = 1 Finding x x + y = 7 x + 1 = 7 x = 6 Solution (6,1) x = 7 – y Check x + y = = 7 7 = 7 3x = 17 + y 3(6) = = 18

Systems Solving Linear Systems Algebraically (Elimination) 5x – 2y = 10 2x + y = 31 5x – 2y = 10 2[2x + y = 31] 5x – 2y = 10 4x + 2y = x + 0y = 72 9x = 72 x = 8 Finding y 2x + y = 31 2(8) + y = y = 31 y = 15 Solution (8, 15) Check 5x – 2y = 10 5(8) – 2(15) = – 30 = = 10 4x + 2y = 62 4(8) + 2(15) = = = 62

Systems Using Systems to Problem Solve A discount movie theater charges $5 for an adult ticket and $2 for a child’s ticket. One Saturday, the theater sold 785 tickets for $3280. How many children’s tickets were sold? Let x = the number of adult tickets Let y = the number of children tickets 5x + 2y = 3280 x + y = 785 5x + 2y = [x + y = 785] 5x + 2y = x – 5y = x – 3y = y = -645 y = 215 Finding x x + y = 785 x = 785 x = adult tickets 215 children tickets

Systems Solving Linear-Quadratic Systems Graphically Two Solutions No Solution One Solution

Systems Solving Linear-Quadratic Systems Graphically y = x 2 – 4x – 2 y = x – 2 m = b = -2 (0,-2) y = x 2 – 4x – 2 x = xy Solutions (0,-2) and (5,3) y = x 2 – 4x – 2 y = x – 2

Systems Solving Systems of Linear Inequalities y < 3x m = 3/1 b = 0 (0,0) y < -2x + 3 m = -2/1 b = 3 (0,3) S y < 3x y -2x + 3 1)Graph each inequality 2)Label each inequality 3)Label the solution region with S 4) Check with calculator or algebraically Solve the system and state one solution. Justify your choice. y < 3x y < -2x + 3 A solution to the system is (1, -3). Justification: y < 3x y < -2x < 3(1) -3 < -2(1) < 3 -3 < True -3 < 1 True

Systems Sarah is selling bracelets and earrings in order to earn money for her summer vacation. She sells bracelets for $2 and earrings for $3. She needs to earn at least $500 and she already knows that she will be selling more than 50 bracelets. Write a system of inequalities to represent the situation. Graph the system. State one possible solution that satisfies the system. State one solution that does not satisfy the system. x: the number of bracelets x > 50 y: the number of earrings 2x + 3y > 500 2x + 3y = 500 2(0) + 3y = 500 3y = 500 y = … y-int: (0, 167) 2x + 3y = 500 2x + 3(0) = 500 2x = 500 x = 250 x-int: (250, 0) Solution: (200 bracelets, 150 earrings) Not a Solution: (100 bracelets, 50 earrings)

Now it’s your turn to review on your own! Using the information presented today and the study guide posted on halgebra.org, complete the practice problem set. Regents Review #5 Tuesday, June 9 th BE THERE!