Linear Inequalities in two Variables

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

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

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

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

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.

Graphing Inequalities in two variables

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.

y ≥ -3/2x + 1 (put into slope intercept to graph easier) 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 >

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!!

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

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 Shade on the side of the line that contains (0,0) -2 2 x

Graph: y < 6

4x – 2y < 7

Rewriting inequalities

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?

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: Which inequality symbol should you use? Read the problem statement carefully Here “$420 or less” Means thet the solution includes, but cannot exceed $420, so use

Rewriting to graph and inequality Solution: Let x = the cost per square foot of the paper Let y = the cost per square foot of tiles. Write an inequality and solve it for y Total cost is $ 420 or less 24x + 12y  420 Subtract 24x from each side 12y  420 - 24x Divide each side by 12 y  35 – 2x y  -2x +35

Rewriting to graph and inequality Solution: Graph y  -2x +35 The inequality symbol is , so the boundary line is solid and you shade below it The graph only make sense in the firts quadrant. Three posible prices per square foot for wall paper and tile are $5 and $25 $5 and $15 $10 and $10

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 = the cost of cashews Let y = the cost of peanuts. Write and inequality and solve it for y Total cost is $ 12 or less 4x + 2y  12 Subtract 4x from each side 2y  12 - 4x Divide each side by 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

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

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

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 x 1 2 3 4 5 6 Graph finding x- intercept and y-intercept x-intecept = 30/10 = 3 y-intercept = 30/5 = 6 Tennisl court use hours 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 Racquetball court use hours

You and some friends have $30 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?

A school fundraiser sells holiday cards and wrapping paper 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.