Graphs of linear inequalities

Graphs of linear inequalities

The graph of a linear inequality in two variables represents all points in the xy-plane that satisfy the inequality. The boundary of this region is a straight line.

Linear inequalities discussed here have the forms:

1. $ax + by \le c$
2. $ax + by \ge c$
3. $ax + by < c$
4. $ax + by > c$

where $a$, $b$, and $c$ are real numbers, and $(a, b) \neq (0,0)$.

Steps for graphing linear inequalities

1. Replace the inequality with an equality: Change the inequality symbol to an equal sign to get the equation of the boundary line.
2. Draw the boundary line:
   • Use a dotted line for strict inequalities (< or >).
   • Use a solid line for inclusive inequalities (≤ or ≥).
3. Choose a test point: Select a convenient point not on the boundary line. The origin (0,0) is often a good choice.
4. Test the inequality: Substitute the coordinates of the test point into the original inequality.
   • If the inequality is true, shade the region containing the test point.
   • If the inequality is false, shade the region opposite the test point.

Examples

Example 11.3: Graph $2x + y \le 4$

Solution:

1. Boundary line: $2x + y = 4$

2. Draw the line: Since the inequality is ≤, use a solid line. Find the intercepts:

   x	y
   0	4
   2	0

3. Test point: Choose (0,0).

4. Test the inequality: $2(0) + 0 \le 4 \Rightarrow 0 \le 4$ (True)

Since the inequality is true for (0,0), shade the region containing the origin (see below).

Example 11.4: Graph $3x + 2y \ge 6$

Solution:

1. Boundary line: $3x + 2y = 6$

2. Draw the line: Use a solid line (≥). Find the intercepts:

   x	y
   0	3
   2	0

3. Test point: Choose (0,0).

4. Test the inequality: $3(0) + 2(0) \ge 6 \Rightarrow 0 \ge 6$ (False)

Since the inequality is false for (0,0), shade the region opposite the origin (see below).

Example 11.5: Graph the system

$$\begin{cases} 2x + y \le 10 \\ 2x + 3y \le 18 \\ x \ge 0 \\ y \ge 0 \end{cases}$$

Solution:

1. Boundary lines: $2x + y = 10$, $2x + 3y = 18$, $x = 0$, $y = 0$

2. Draw the lines: Use solid lines (≤, ≥). Find intercepts:

   Equation	x-intercept	y-intercept
   2x + y = 10	5	10
   2x + 3y = 18	9	6

3. Test point: Choose (0,0).

4. Test the inequalities:

   • $2(0) + 0 \le 10 \Rightarrow 0 \le 10$ (True)
   • $2(0) + 3(0) \le 18 \Rightarrow 0 \le 18$ (True)
   • $0 \ge 0$ (True)
   • $0 \ge 0$ (True)

Shade the region where all inequalities are true. This will be the region bounded by the lines in the first quadrant (because of $x \ge 0$ and $y \ge 0$).