1. Regents Review #4 Inequalities and Systems

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

3. Simple Inequalities -3x – 4 > 8 -3x > 12 x < - 4 3(2x – 1) + 3x 4(2x + 1) 6x – 3 + 3x 8x + 4 9x – 3 8x + 4 x 7 -7 -6 -5 -4 -3 -2 -1 4 5 6 7 8 9 10

4. 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.

5. 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 “OR” Graph the combination of both solutions sets

6. Compound Inequalities “AND” -12 2x < -8 2x -12 and 2x < -8 x -6 and x < -4 -6 x < - 4 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -3 < x 4 x > -3 and x 4 -4 -3 -2 -1 0 1 2 3 4 5 6

7. Compound Inequalities “OR” x < -4 or x 6 -10 -8 -6 -4 -2 0 2 4 6 8 10 2x + 5 15 2x 5 x < 3 x 5 0 1 2 3 4 5 6 7 8 9 10 11

8. Linear Inequalities Graph Linear Inequalities the same way you graph Linear Equations 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

9. Linear Inequalities Graph -2y > 2x – 4 -2y > 2x – 4 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

10. 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

11. 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

12. Systems y = 4x – 1 m = b = -1 (0,-1) 3x + 2y = 20 2y = -3x + 20 y = - + 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 – 1 6 + 14 = 20 7 = 7 20 = 20 Solve the system y = 4x – 1 and 3x + 2y = 20 graphically

13. 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 6 + 1 = 7 7 = 7 3x = 17 + y 3(6) = 17 + 1 18 = 18

14. Systems Solving Linear Systems Algebraically (Elimination) 5x – 2y = 10 2x + y = 31 5x – 2y = 10 2[2x + y = 31] 5x – 2y = 10 4x + 2y = 62 + 9x + 0y = 72 9x = 72 x = 8 Finding y 2x + y = 31 2(8) + y = 31 16 + y = 31 y = 15 Solution (8, 15) Check 5x – 2y = 10 5(8) – 2(15) = 10 40 – 30 = 10 10 = 10 4x + 2y = 62 4(8) + 2(15) = 62 32 + 30 = 62 62 = 62

15. Systems Using Systems to Solve Word Problems 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 = 3280 -5[x + y = 785] 5x + 2y = 3280 -5x – 5y = -3925 + 0x – 3y = -645 -3y = -645 y = 215 Finding x x + y = 785 x + 215 = 785 x = 570 570 adult tickets 215 children tickets

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

17. 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 3 0-2 1-5 2-6 3-5 4-2 53 Solutions (0,-2) and (5,3) y = x 2 – 4x – 2 y = x – 2

18. 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 Solve the system: y < 3x y -2x + 3

19. Now it’s your turn to review on your own! Using the information presented today and your review packet, complete the practice problems in the packet. Regents Review #5 is FRIDAY, May 31 st BE THERE!!!!