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Systems of Linear InequalitiesSystem Connection between two (or more)
functions where the inputs and theoutputs are the same
Linear Inequality Shaded graph that shows the possible solutions/ordered pairs that make the inequality true
The solution set to a system of linear inequalities will be the shaded area representing all of the ordered pairs that makes BOTH inequalities true.
y > 2x y > x 3yint: (0,0) yint: (0,3)slope: 2/1 or 2/1 slope: 1/1 or 1/1solid line (included) dotted line (not included)test: (0,2) 2 > 0 test: (0,0) 0 > 3
A The area is not part of either solution set.
B The solution set for the yellow (dottedline) inequality, but not the
other inequality.
C The solution set for the blue (solidline) inequality, but not the other inequality
D The solution set for both inequalities
Is the ordered pair part of the solution set or not?
A yes B yes C no ( yes, no)D no ( yes, no)E no F no G no H yes I no J no
AH
BG
D I
FJ
E
C
Graphing Systems of Inequalities1. Get first inequality in slopeintercept form. Remember: if you multiply or divide by a negative, flip the inequality sign
2. Graph the points of the first inequality. Plot the yintercept (0, b) Write both forms of slope (rise/run) Graph all possible points
3. Look at the first inequality sign to see if the points that were graphed are included or not included in the solution set. Solid line if included (< or >) Dotted/Dashed line if not included (< or >)
4. For the first inequality, shade the side that has the solution set where the ordered pairs make the linear inequality true. Test Point...(0,0) unless on the line
5. Repeat Step 14 for the second inequality. Convert to slopeintercept form Plot the points Solid or dotted/dashed line Shade (Test Point)
6. Darken the area shared by both inequalitiesThe solution set is ONLY the part shared by both inequalities.
2x + 3y > 64x 2y > 4
2x + 3y > 6 4x 2y > 4
y > x + 2yintercept: rise/run: type of line: solid or dottedtest point: (__,__)
y < 2x + 2yintercept: rise/run: type of line: solid or dottedtest point: (__,__)
23
12
Graph the solution set.
y > 1x < 2y < x + 1yintercept: (0,1)
rise/run: 1/2 or 1/2type of line: solid or dottedtest point: (__,__)
0 < .5(0) + 1 0 < 1true, so (0,0) is included in the solution set
0 0
Thought Process for Systems of Linear Inequalities M/CIf the lines are same on each graph...
Choose an ordered pair in each of the possible shaded areas and test them with both inequalities...the correctly shaded graph will make both linear inequalities true.
If the lines are not the same on each graph...
First, identify which graph(s) are correct by looking at the type of line/yintercept/rise over run. Then, choose a test point to see which shaded area shows the solution set...the correctly shaded graph will make both linear inequalities true
Is (2,3) a possible solution to the system of inequalities?
2x 3y < 44x + y > 10
2(2) 3(3) < 44 + 9 < 413 < 4
False, not part of the solution set of the first inequality, so not part of the solution set of the system of inequalities.
Is (2,3) a possible solution to the system of inequalities?
x + 3y < 44x y > 10
2 + 3(3) < 4 4(2) (3) > 102 9 < 4 8 + 3 > 10 7 < 4 11 > 10
Yes, it is a possible solution.
Which graph represents the following system of inequalities?y < x + 1x y < 2
(A) (B) (C) (D)
23
Which graph represents the following system of inequalities?x y < 13x + y > 2
(A) (B) (C) (D)
Which area should be shaded to represent the solution set to the following system of linear inequalities?
A
B
C
D
y < 4x + 3y < x 2
A test point (0,4)4 < 4(0) + 3 4 < 3 falseA cannot be the answer
B test point (0,0)0 < 4(0) + 3 0 < 3 true0 < (0) 2 0 < 2 false
C test point (0,3)3 < 4(0) + 3 3 < 3 true3 < (0) 2 3 < 2 true
A represents the solution set because the ordered pairs make both inequalities true.
Which area should be shaded to represent the solution set to the following system of linear inequalities?
A
B
C
D
4x + y < 2x y < 3
A test point (0,0)4(0) + (0) < 2 0 < 2 true(0) (0) < 3 0 < 3 trueA represents the solution set because the ordered pairs make both inequalities true.
Systems of Linear Inequalities Word Problems Remember the common types of systems relationships.
1. Two variables added to get a total. 2. Two variables times a rate to get a total.3. Two scenarios given.4. Relationship between the two variables.
*Also could be the minimums/maximums for each variable.
Phrases that will indicate an inequality:"a is more than b" means a > b"a is at least b" means a > b"a is less than b" means a < b"a is no more than b" means a < b
Steps to be Successful:1. What do you not know/what are you looking for?2. Assign variables to what you don't know/what you are looking for. (Write it down!)3. Identify other information/numbers in the problem.
See the quantities and their relationship instead of numbers.
John is doing a fundraiser for school. He needs to sell at least $200 worth of items. Shirts cost $10 each and hats cost $8 each. He must sell more than 12 hats. Write a system of linear inequalities to model this situation.
x = # of shirts 10x + 8y > 200y = # of hats y >12
You can work a maximum of 40 hours a week. You need to make at least $400 in order to cover your expenses. Your office job pays $12 an hour and your babysitting job pays $10 an hour. Write a system of linear inequalities to model this situation.
x = # of office hoursy = # of babysitting hours
12x + 10y > 400x + y < 40
Systems of Linear Inequalities Word Problems Remember the common types of systems relationships.
1. Two scenarios given in the problem.2. Two variables added to get a total.3. Two variables (# of) times a rate to get a total.4. Two variables (rate) times a number to get a total.5. Rate is given in the problem but the total is not.6. A relationship between the two variables. *Also could be the minimums/maximums for each variable.
Phrases that will indicate an inequality:"... more than the total" means amount > total"... at least the total" means amount > total"... less than the total" means amount < total"... no more than the total" means amount < total
Steps to be Successful:1. What do you not know/what are you looking for?2. Assign variables to what you don't know/what you are looking for. (Write it down!)3. Identify other information/numbers in the problem.
See the quantities and their relationship instead of numbers.
John is doing a fundraiser for school. He needs to sell at least $200 worth of items. Shirts cost $10 each and hats cost $8 each. He must sell more than 12 hats. Write a system of linear inequalities to model this situation.
x = # of shirtsy = # of hats
10x + 8y > 200y > 12
You can work a maximum of 40 hours a week. You need to make at least $400 in order to cover your expenses. Your office job pays $12 an hour and your babysitting job pays $10 an hour. Write a system of linear inequalities to model this situation.
x = # of office hoursy = # of babysitting hours
12x + 10y > 400x + y < 40