2Linear InequalitiesA 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
3Linear InequalitiesA 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
4Linear InequalitiesA solution of an inequality is an ordered pair that make the inequality trueAn ordered pair (x,y) is a solution if it makes the inequality true
5Checking Solutions 3x + 2y ≥ 2 Are the following solutions to: (0,0) (2,-1) (0,2)3(0) + 2(2) ≥ 24 ≥ 2Is a solution3(0) + 2(0) ≥ 20 ≥ 2Not a solution3(2) + 2(-1) ≥ 24 ≥ 2Is a solution
6Graphing 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.
8To 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 solutionYou 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.
9y ≥ -3/2x + 1 (put into slope intercept to graph easier) The graph of an inequality is the graph of all the solutions of the inequality3x+ 2y ≥ 2y ≥ -3/2x + 1 (put into slope intercept to graph easier)Graph the line that is the boundary of 2 half planesBefore you connect the dots check to see if the line should be solid or dashedsolid if ≥ or ≤dashed if < or >
10y ≥ -3/2x + 1 Step 1: graph the boundary (the line is solid ≥) Step 2: testa point NOTOn the line(0,0) is alwaysThe easiest if it’sNot on the line!!3(0) + 2(0) ≥ 20 ≥ 2Not a solutionSo shade the other side of the line!!
11Graphing Inequalities in one variable An inequality in one variable can be graphed on an number line or in the coordinate planeThe boundary line will be a horizontal or vertical lineX -1Graph x=-1 using dashed lineUse (0,0) as a test point
12Graphing Inequalities in one variable yX -1yGraph x=-1 using dashed lineUse (0,0) as a test pointX -10 -1Shade on the side of the line that contains (0,0)-22x
17Rewriting 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?
18Rewriting 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 carefullyHere “$420 or less”Means thet the solution includes, but cannot exceed $420, so use
19Rewriting to graph and inequality Solution:Let x = the cost per square foot of the paperLet y = the cost per square foot of tiles.Write an inequality and solve it for yTotal cost is $ 420 or less x + 12y 420Subtract 24x from each side y xDivide each side by y 35 – 2xy -2x +35
20Rewriting to graph and inequality Solution:Graph y -2x +35The inequality symbol is , so the boundary line is solid and you shade below itThe graph only make sense in the firts quadrant.Three posible prices per squarefoot for wall paper and tile are$5 and $25$5 and $15$10 and $10
21Rewriting 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 cashewsLet y = the cost of peanuts.Write and inequality and solve it for yTotal cost is $ 12 or less x + 2y 12Subtract 4x from each side 2y xDivide each side by y 6 – 2xy -2x + 6Three posible combination of peanuts and cashews are0 lb of peanuts and 3 lb of cashews3 lb of peanuts and 6 lb of cashews1 lb of peanuts and 1 lb of cashews
22Rewriting an inequality from a graph Which inequality represents the graph at the righ?The slope of the line is 2The y-intercept is 1So the equation of the boundary line isy = 2x+1The symbol is because the region below the boundary is shadedThe correct answer is y 2x+1
24Rewriting 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 courtLet y = the number of hour using the tennis court.Write an inequality 10 x + 5y 30yx123456Graph finding x- intercept and y-interceptx-intecept = 30/10 = 3y-intercept = 30/5 = 6Tennisl courtuse hoursThree possible combinations are:1 hour of racquetball and 2 hours of tennis1 hour of racquetball and 3 hours of tennis2 hours of racquetball and 1 hour of tennisRacquetball courtuse hours
26You 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?
27A 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.