Graphical Solution of Inequalities

[MathDefinition title="Graphical Solution"] The graphical solution of an inequality in two variables is the region of the coordinate plane that satisfies the inequality. The boundary line divides the plane into two half-planes. [/MathDefinition]

[MathDefinition title="Rules for Graphing Inequalities"]

1. For ax + by < c or ax + by > c: Draw a dashed line (boundary not included)
2. For ax + by ≤ c or ax + by ≥ c: Draw a solid line (boundary included)
3. Test a point (usually origin) to determine which side to shade [/MathDefinition]

[MathKeyFormula title="Key Concepts"]

1. Linear inequality: ax + by ≤ c or ax + by ≥ c
2. The solution region is a half-plane
3. For strict inequalities (< or >), use dashed line
4. For non-strict inequalities (≤ or ≥), use solid line
5. The intersection of multiple inequalities gives the feasible region [/MathKeyFormula]

[MathNote title="Graphing Steps"]

1. Convert inequality to equation to find the boundary line
2. Plot the boundary line
3. Choose a test point (0,0) if possible
4. Substitute the test point into the inequality
5. If true, shade the side containing the test point
6. If false, shade the opposite side [/MathNote]

[MathExample title="Example 1: Simple Inequality"] Problem: Solve the inequality 2x + 3y ≤ 6 graphically.

Solution: Step 1: Convert to equation: 2x + 3y = 6

Step 2: Find intercepts: x-intercept: y = 0, x = 3 → (3, 0) y-intercept: x = 0, y = 2 → (0, 2)

Step 3: Draw a solid line through (3, 0) and (0, 2)

Step 4: Test point (0, 0): 2(0) + 3(0) = 0 ≤ 6 ✓

Since the test point satisfies the inequality, shade the region containing (0, 0).

The solution is the region below and including the line 2x + 3y = 6. [/MathExample]

[MathExample title="Example 2: System of Inequalities"] Problem: Solve the system: x + y ≤ 4, x ≥ 0, y ≥ 0 graphically.

Solution: Step 1: Graph x + y ≤ 4 Boundary: x + y = 4 Intercepts: (4, 0) and (0, 4) Solid line, shade below (test (0,0): 0 ≤ 4 ✓)

Step 2: Graph x ≥ 0 This is the region to the right of and including the y-axis

Step 3: Graph y ≥ 0 This is the region above and including the x-axis

Step 4: Find the intersection The feasible region is a triangle with vertices at (0, 0), (4, 0), and (0, 4) [/MathExample]

[MathExample title="Example 3: Strict Inequality"] Problem: Solve the inequality x − 2y > 4 graphically.

Solution: Step 1: Convert to equation: x − 2y = 4

Step 2: Find intercepts: x-intercept: y = 0, x = 4 → (4, 0) y-intercept: x = 0, y = −2 → (0, −2)

Step 3: Draw a dashed line through (4, 0) and (0, −2)

Step 4: Test point (0, 0): 0 − 2(0) = 0 > 4? No, 0 is not greater than 4

Since (0, 0) does not satisfy the inequality, shade the opposite side (the side not containing the origin).

The solution is the region above and to the right of the dashed line x − 2y = 4. [/MathExample]

[MathExample title="Example 4: Quadratic Inequality"] Problem: Solve the inequality y ≥ x² − 4 graphically.

Solution: Step 1: Convert to equation: y = x² − 4

Step 2: This is a parabola opening upward with vertex at (0, −4)

Step 3: Draw a solid parabola

Step 4: Test point (0, 0): 0 ≥ 0² − 4 0 ≥ −4 ✓

Since the test point satisfies the inequality, shade the region inside the parabola (above the curve).

The solution is the region on and above the parabola y = x² − 4. [/MathExample]