Section 3.2 Graphing Systems of Linear Inequalities in Two Variables

Procedure for Graphing Linear Inequalities:

1. Draw the graph of the equation obtained for the given inequality by replacing the inequality

sign with an equal sign. Use a dashed line if the problem involves a strict inequality, < or >.

Otherwise, use a solid line to indicate that the line itself constitutes part of the solution.

2. Pick a test point, (a, b), lying in one of the half-planes determined by the line sketched in Step

1 and substitute the numbers a and b for the values of x and y in the given inequality. For

simplicity, use the origin, (0, 0), whenever possible.

3. If the inequality is satisfied (True), the graph of the solution to the inequality is the half-plane

containing the test point (Shade the region containing the test point). Otherwise (if the inequality

is False), the solution is the half-plane not containing the test point (Shade the region that does

not contain the test point).

1. Find the graphical solution of the inequality. Does the solution exist on the line 10x+ 4y = �8?

10x+ 4y �8

2. Find the graphical solution of the inequality. Does the solution exist on the line 2x+ 5y = �10?

2x+ 5y > �10

3. Write a system of linear inequalities that describes the shaded region.

5x+ 2y 30

x+ 2y 12

x 0

y 0

Bounded vs Unbounded A solution set S is bounded if the solution has a boundary that

fully encloses the solution set.

A solution set S is unbounded if the solution DOES NOT have a boundary that fully encloses

the solution set.

4. Determine graphically the solution set for the system of inequalities. Indicate whether the solution

set is bounded or unbounded.

3x� 2y > �17

�x+ 2y > 7

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5. Determine graphically the solution set for the system of inequalities. Indicate whether the solution

set is bounded or unbounded.

3x� 4y 12

4x+ 5y 20

x � 0

y � 0

6. Determine graphically the solution set for the system of inequalities. Indicate whether the solution

set is bounded or unbounded.

2x+ 2y � 20

6x+ 6y 30

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set is bounded or unbounded.

x+ y 5

2x+ y 8

2x� y � �1

x � 0

y � 0

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