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Systems of Linear Inequalities in Two Variable

Outline

1 Graphing inequalities

2 Word problems and inequalities

3 Graphing systems of inequalities

Problem
Graph the following inequalities:

3x + 4y ≤ 24

3x + 4y > 12.

The method

Follow these steps to graph the inequality 3x + 4y ≤ 24.

Graph the line 3x + 4y = 24; the intercept method is best for this
purpose.

• The plane has been divided into two halves , only one of which
corresponds to the inequality .

• Determine which half to shade by analyzing a test point. Since (0, 0)
satisfies the inequality , it is in the half to be shaded and we shade
accordingly.

• You can see these steps on the next slide.

Example

Problem

Fruit Punch A is 10% lemon juice. Fruit Punch B is 20% lemon juice. If
you require at least 30 ounces of lemon juice each day, find a system of
inequalities that describes those combinations of punch which satisfy your
requirements.

Solution

Let a and b denote the number of ounces of punch A and B. Then

Problem
Graph the system of inequalities: 5x + 2y ≤ 20, x ≥ 0, and y ≥ 0. In
other words, find all points in the plane that simultaneously satisfy the set
of inequalities .


Problem

Graph the system of inequalities :

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