Inequalities

Inequalities

Forming Linear Inequalities in Two Unknowns from Word Problems

Linear Inequalities

In the xy-plane, any straight line divides the plane into two distinct regions, called half-planes.

For example, consider the equation $y = 5$ drawn on the xy-plane. This line separates the plane into two half-planes:

• Points above the line (in half-plane A) satisfy the inequality $y > 5$.
• Points below the line (in half-plane B) satisfy the inequality $y < 5$.

Shading of Regions

In linear programming, the region of interest is usually left clear, and the unwanted regions are shaded. For example, when drawing the inequality $y > 4$, follow these steps:

1. Draw the line $y = 4$ to separate the plane into $y > 4$ (above the line) and $y < 4$ (below the line).
2. Shade the unwanted region (below the line, in this case), leaving the region that satisfies $y > 4$ clear.

Example 1

Draw and show the half-plane represented by the inequality: $8x + 2y \geq 16$

Solution:

1. First, graph the line $8x + 2y = 16$, which is the boundary line.
2. The inequality $8x + 2y \geq 16$ means we want the region above or on the line. So, test a point not on the line (e.g., the origin (0,0)) to determine which side of the line satisfies the inequality. Testing (0,0):
   • $8(0) + 2(0) = 0$ (which is less than 16, so this region is not valid)
   • Therefore, shade the region above the line since it satisfies $8x + 2y \geq 16$.

Feasible Region

The feasible region is the region where all the inequalities in a linear programming problem intersect. To clearly show the feasible region, the unwanted regions of the inequalities must be shaded, leaving the required region clear.

Example 2

Show the feasible region that satisfies the following inequalities: $3x + 3y > 12$ and $y - x < 2$

Solution:

1. For $3x + 3y > 12$:
   • First, graph the boundary line $3x + 3y = 12$.
   • Test a point not on the line, such as the origin (0,0):
     • $3(0) + 3(0) = 0$ (which is less than 12, so the region below the line does not satisfy the inequality).
   • Therefore, shade the region above the line.

1. For $y - x < 2$:
   • Graph the boundary line $y - x = 2$.
   • Test the origin (0,0):
     • $0 - 0 = 0$ (which is less than 2, so the region below the line satisfies the inequality).
   • Therefore, shade the region above the line.

After plotting both inequalities, the feasible region is the intersection of the regions that satisfy both inequalities. You should shade the unwanted regions, leaving the feasible region clear.