L inear Inequalities in two Variables

1. Linear Inequalities in two Variables

2. Linear Inequalities A linear inequality in two variables such as y  x-2, can be formed by replacing the equals sign in a linear equation with an inequality symbol

3. Linear Inequalities A linear inequality in two variables has an infinity number of solutions. These solutions can be represented in the coordinate plane as the set of all points on one side of a boundary line

4. Linear Inequalities • A solution of an inequality is an ordered pair that make the inequality true • An ordered pair (x,y) is a solution if it makes the inequality true

5. Checking Solutions Are the following solutions to: 3x + 2y ≥ 2 (0,0) (2,-1) (0,2) 3(0) + 2(2) ≥ 2 4 ≥ 2 Is a solution 3(2) + 2(-1) ≥ 2 4 ≥ 2 Is a solution 3(0) + 2(0) ≥ 2 0 ≥ 2 Not a solution

6. Graphing Inequalities in two variables • The graph of two inequalities in two variables consists in all points in the coordinate plane that represent the solution. • The graph is the region called half-plane that is bounded by a line.

7. Graphing Inequalities in two variables

8. To sketch the graph of a linear inequality: • Sketch the line given by the corresponding equation (solid if ≥ or ≤, dashed if < or >). This line separates the coordinate plane into 2 half-planes. • In one half-plane – all of the points are solutions of the inequality. In the other half-plane - no point is a solution • You can decide whether the points in an entire half-plane satisfy the inequality by testing ONE point in the half-plane. • Shade the half-plane that has the solutions to the inequality.

9. The graph of an inequality is the graph of all the solutions of the inequality 3x+ 2y ≥ 2 y ≥ -3/2x + 1 (put into slope intercept to graph easier) Graph the line that is the boundary of 2 half planes Before you connect the dots check to see if the line should be solid or dashed solid if ≥ or ≤ dashed if < or >

10. y ≥ -3/2x + 1 Step 1: graph the boundary (the line is solid ≥) Step 2: test a point NOT On the line (0,0) is always The easiest if it’s Not on the line!! 3(0) + 2(0) ≥ 2 0 ≥ 2 Not a solution So shade the other side of the line!!

11. Graphing Inequalities in one variable • An inequality in one variable can be graphed on an number line or in the coordinate plane • The boundary line will be a horizontal or vertical line X -1 • Graph x=-1 using dashed line • Use (0,0) as a test point

12. Graphing Inequalities in one variable y X -1 y • Graph x=-1 using dashed line • Use (0,0) as a test point X  -1 0  -1 -2 • Shade on the side of the line that contains (0,0) 2 x

13. Graph: y < 6

14. 4x – 2y < 7

15. Rewriting inequalities

17. Rewriting to graph and inequality An interior decorator is going to remodel a kitchen. The wall above the stove and the counter is going to be redone as shown. The owners can spend $ 420 or less. Write a linear inequality and graph the solutions. What are the three possible prices for the wall paper and tiles?

18. Rewriting to graph and inequality An interior decorator is going to remodel a kitchen. The wall above the stove and the counter is going to be redone as shown. The owners can spend $ 420 or less. Write a linear inequality and graph the solutions. What are the three possible prices for the wall paper and tiles? Solution: Whichinequality symbol shouldyou use? Readtheproblemstatementcarefully Here “$420 orless” Meansthetthesolutionincludes, butcannotexceed $420, so use

19. Rewriting to graph and inequality Solution: Let x = thecost per squarefoot of thepaper Let y = thecost per squarefoot of tiles. Writeaninequality and solveitfor y Total costis $ 420 orless 24x + 12y  420 Subtract 24x fromeachside 12y  420 - 24x Divide eachsideby 12 y  35 – 2x y  -2x +35

20. Rewriting to graph and inequality Solution: Graph y  -2x +35 Theinequality symbol is, so theboundary line issolid and youshadebelowit Thegraphonlymakesense in thefirtsquadrant. Three posible prices per square footforwallpaper and tile are $5 and $25 $5 and $15 $10 and $10

21. Rewriting to graph and inequality Solve: For a party, you can spend no more than $12 on nuts. Peanuts cost $2/lb. Cashews cost $4/lb. What are the three possible combination of peanut and cashews you can buy? Let x = thecost of cashews Let y = thecost of peanuts. Write and inequality and solveitfor y Total costis $ 12 orless 4x + 2y  12 Subtract 4x fromeachside 2y  12 - 4x Divide eachsideby 2 y  6 – 2x y  -2x + 6 Three posible combination of peanuts and cashews are 0 lb of peanuts and 3 lb of cashews 3 lb of peanuts and 6 lb of cashews 1 lb of peanuts and 1 lb of cashews

22. Rewriting an inequality from a graph Which inequality represents the graph at the righ? The slope of the line is 2 The y-intercept is 1 So the equation of the boundary line is y = 2x+1 The symbol is  because the region below the boundary is shaded The correct answer is y  2x+1

23. Activities Skill practice p. 71 1-7 p.72 10 and 11 Homework p. 72 1-9

24. Rewriting to graph and inequality Solve: You and a friend can spend no more than $30 at a health club. It cost $10 an hour to use the racquetball court and $5 an hour to use the tennis court. Find three possible combinations of racquetball court and tennis court times you and your friend can use? Let x = the number of hour using the racquetball court Let y = the number of hour using the tennis court. Write an inequality 10 x + 5y  30 y Graph finding x- intercept and y-intercept x-intecept = 30/10 = 3 4 y-intercept = 30/5 = 6 Tennislcourt use hours 3 Three possible combinations are: 1 hour of racquetball and 2 hours of tennis 1 hour of racquetball and 3 hours of tennis 2 hours of racquetball and 1 hour of tennis 2 1 x 1 2 3 4 5 6 Racquetballcourt use hours

26. You and some friends have $30. You want to order large pizzas (p) that are $9 each and drinks (d) that cost $1 each. Write and graph an inequality that shows how many pizzas and drinks can you order.? Tickets to a play cost $5 at the door and $4 in advance. The theater club wants to raise at least $400 from the play. Write and graph an inequality for the number of tickets the theater club needs to sell. If the club sells 40 tickets in advance, how many do they need to sell at the door to reach their goal?

27. A school fundraiser sells holiday cards and wrapping paper. They are trying to raise at least $400. They make a profit of $1.50 on each box of holiday cards and $1.00 on each pack of wrapping paper. a. What is an inequality for the profit the school wants to make for the fundraiser? b. If the fundraiser sells 100 boxes of cards and 160 packs of wrapping paper, will they reach their goal? Show your work.