Embed Size (px)
Citation preview
ALGEBRA 1
Lesson 6-6 Warm-Up
ALGEBRA 1
“Systems of Linear Inequalities” (6-6)
What is a “system of linear inequalities”?
What is a “solution of system of linear inequalities”?
system of linear inequalities : two or more linear inequalities on a graph
Solution of a System of Linear Inequalities : a solution that makes all of the inequalities in a system true
In the graph below, the lavendar-shaded area include all of the solutions of the system represented by x ≥ 3 and y -2 (in other words, it include all of the solutions that make both inequalities true).
ALGEBRA 1
“Systems of Linear Inequalities” (6-6)
What is a “solution set”?
Solution Set: the graph that shows the “set” of all solutions to the system
Example: What is the solution set of y 2x – 5 and 3x + 4y 12 .
The points where the two inequalities overlap shaded in lavender is the solution set to the system.
ALGEBRA 1
“Systems of Linear Inequalities” (6-6)
How can you describe the four quadrants of a coordinate grid with systems of linear inequalities”?
ALGEBRA 1
Solve by graphing. y < –x + 3
–2x + 4y 0
Check: The point (–1, 1) is in the region graphed by both inequalities. See if (–1, 1) satisfies both inequalities.
>–
Graph y = –x + 3 and –2x + 4y = 0.
Systems of Linear InequalitiesLESSON 6-6
Additional Examples
–2x + 4y = 0
+2x +2x
4y = 2x__ __ 1y = x + 0
1
212
ALGEBRA 1
(continued)
The coordinates of the points in the [light blue] region where the graphs of the two inequalities overlap are solutions of the system.
y < –x + 3 –2x + 4y 0
1 < –(–1) + 3 Substitute (–1, 1) for (x, y). –2(–1) + 4(1) 0
1 < 4 2 + 4 0
>–
>–
>–
Systems of Linear InequalitiesLESSON 6-6
Additional Examples
ALGEBRA 1
Write a system of inequalities for each shaded region
below.
System for the [light blue] region: y < – x + 2
y < 4
12
[red] region
boundary: y = – x + 2and y = 4
12
The region lies between the boundary lines, so the inequality is
y < – x + 2 and y 412
[blue] region
Boundary lines: y = 4 and y = – x + 2
The region lies below the boundary line, so the inequality is
y – x + 2 and y < 4.
Systems of Linear InequalitiesLESSON 6-6
Additional Examples
12
12
ALGEBRA 1
You need to make a fence for a dog run. The length of
the run can be no more than 60 ft, and you have 136 feet of
fencing that you can use. What are the possible dimensions of
the dog run?
Define: Let = length of the dog run.
Let w = width of the dog run.
Words : The is no 60 ft. The is no 136
ft.length more than perimeter more than
Equation: 60 2 + 2 w 136
<– <–
Solve by graphing. 60
2 + 2w 136
<–<–
Systems of Linear InequalitiesLESSON 6-6
Additional Examples
ALGEBRA 1
(continued)
The solutions are the coordinates of the points that lie in the region shaded light-blue and on the lines = 60 and 2 + 2w = 136.
60m = 0: b = 60
Shade below = 60.
<–
Test (0, 0). 2(0) + 2(0) 136
0 136
So shade below
2 + 2w = 136
<–<–
2 + 2w 136
Graph the intercepts (68, 0) and (0, 68).
<–
Systems of Linear InequalitiesLESSON 6-6
Additional Examples
ALGEBRA 1
Suppose you have two jobs, babysitting, which pays $5
per hour and sacking groceries, which pays $6 per hour. You can
work no more than 20 hours each week, but you need to earn at
least $90 per week. How many hours can you work at each job?
Define: Let b = hours of babysitting.
Let s = hours of sacking groceries.
Words: The number is less 20. The amount is at 90.of hours than or earned leastworked equal to
Equation: b + s 20 5 b + 6 s 90>–<–
Systems of Linear InequalitiesLESSON 6-6
Additional Examples
ALGEBRA 1
(continued)
The solutions are all the coordinates of the points that are nonnegative integers that lie in the region shaded light-blue and on the lines b + s = 20 and 5b + 6s = 90.
Solve by graphing. b + s 205b + 6s 90>–
<–
Systems of Linear InequalitiesLESSON 6-6
Additional Examples
ALGEBRA 1
Solve each system by graphing.
1. x 0 2. 2x + 3y > 12 3. y x – 3
y < 3 2x + 2y < 12 2x – 3y –9
>–23
>–
>–
Systems of Linear InequalitiesLESSON 6-6
Lesson Quiz
ALGEBRA 1
4. Write a system of inequalities for the following graph.
y < x + 3
y > – x – 212
Systems of Linear InequalitiesLESSON 6-6
Lesson Quiz
