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 SolutionsAre the following solutions to:

3x + 2y ≥ 2

(0,0) (2,-1) (0,2)

3(0) + 2(0) ≥ 2 0 ≥ 2Not a solution

3(2) + 2(-1) ≥ 2 4 ≥ 2 Is a solution

3(0) + 2(2) ≥ 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:

1. Sketch the line given by the corresponding equation (solid if ≥ or ≤, dashed if < or >). This line separates the coordinate plane into 2 half-planes.

2. In one half-plane – all of the points are solutions of the inequality. In the other half-plane - no point is a solution

3. You can decide whether the points in an entire half-plane satisfy the inequality by testing ONE point in the half-plane.

4. Shade the half-plane that has the solutions to the inequality.

The graph of an inequality is the graph of all the solutions of the inequality

3x+ 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 dashed solid if ≥ or ≤ dashed if < or >

y ≥ -3/2x + 1Step 1: graphthe 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) ≥ 2 0 ≥ 2Not 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

X -1 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)

y

x-2 2

y

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 inequalitySolution:

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 inequalitySolution:

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 + 6Three posible combination of peanuts and cashews are

0 lb of peanuts and 3 lb of cashews 3 lb of peanuts and 6 lb of cashews1 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 1So 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 inequalitySolve: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

Graph finding x- intercept and y-intercept

x-intecept = 30/10 = 3

y-intercept = 30/5 = 6

y

x1

2

3

4

1 2 3 4 5 6

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 courtuse hours

Ten

nis

l co

urt

use

ho

urs

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. 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.