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SYSTEMS OF LINEAR INEQUALITIES
Solving Linear Systems of Inequalities by Graphing
Solving Systems of Linear Inequalities
1. We show the solution to a system of linear inequalities by graphing them.
a) This process is easier if we put the inequalities into Slope-Intercept Form, y = mx + b.
Solving Systems of Linear Inequalities
2. Graph the line using the y-intercept & slope.
a) If the inequality is < or >, make the lines dotted.
b) If the inequality is < or >, make the lines solid.
Solving Systems of Linear Inequalities
3. The solution also includes points not on the line, so you need to shade the region of the graph:
a) above the line for ‘y >’ or ‘y ’.b) below the line for ‘y <’ or ‘y ≤’.
Solving Systems of Linear Inequalities
Example: a: 3x + 4y > - 4
b: x + 2y < 2
Put in Slope-Intercept Form: ) 3 4 4
4 3 4
31
4
a x y
y x
y x
) 2 2
2 2
11
2
b x y
y x
y x
Solving Systems of Linear Inequalities
a:
dotted
shade above
b:
dotted
shade below
Graph each line, make dotted or solid and shade the correct area.
Example, continued:
3: 1
4 a y x 1
: 12
b y x
Solving Systems of Linear Inequalities
a: 3x + 4y > - 4
3: 1
4 a y x
Solving Systems of Linear Inequalities
a: 3x + 4y > - 4
b: x + 2y < 2
3: 1
4 a y x
1: 1
2 b y x
Solving Systems of Linear Inequalities
a: 3x + 4y > - 4
b: x + 2y < 2
The area between the green arrows is the region of overlap and thus the solution.
2 2
1
y x
y x
2
2 1
x y
y x
The dry cleaner charges $4 to clean a pair of pants and $3 to clean a shirt. You want to get at least 8 items cleaned. You have $32 to spend on dry cleaning.a. Write a system of inequalities to model the situation.
Suppose you are buying two kinds of notebooks for school. A spiral notebook costs $2, and a three-ring notebook costs $5. You must have at least 6 notebooks. The cost of the notebooks can be no more than $20.a. Write a system of inequalities to model the situation.b. Graph and solve the system.
A camp counselor needs no more than 30 campers to sign up for two mountain hikes. The counselor needs at least 10 campers on the low trail and at least 5 campers on the high trail.a. Write a system of inequalities to model the situation.b. Graph and solve the system.
