A linear inequality involves a linear equation, such as: $\ y > 2x + 1$

Steps to Graph:

1. Graph the Boundary Line:

• Start by graphing the equation $\ y = 2x + 1$ as a straight line.
• If the inequality is strict $> \ or \ <$, use a dashed line to indicate that points on the line are not included.
• If the inequality is non-strict $\geq$ or $\ \leq$, use a solid line to show that points on the line are included.

2. Determine the Shaded Region:

• Choose a test point (often the origin, $\ (0, 0)$ to determine which side of the line to shade.
• Substitute the test point into the inequality. If the inequality holds true, shade the region that includes the test point; otherwise, shade the opposite side.

Example: For $\ y > 2x + 1$:

• Draw the line $y = 2x + 1$ with a dashed line.
• Test $(0, 0) : 0 > 2(0) + 1$ is false, so shade the region above the line.

Quadratic inequalities involve a quadratic equation, such as: $y \leq x^2 - 4$

Steps to Graph:

3. Graph the Parabola:

• Start by graphing the equation $y = x^2 - 4 .$
• Use a solid line for $\ \leq \ or \ \geq$, and a dashed line for $\ < or \ > .$

4. Determine the Shaded Region:

• Similar to linear inequalities, choose a test point to determine which side of the parabola satisfies the inequality.
• For quadratic inequalities like $\ y \leq x^2 - 4 ,$ the region below the parabola is typically shaded.

Example: For$\ y \leq x^2 - 4$:

• Graph the parabola $\ y = x^2 - 4$ using a solid line.
• Test $\ (0, 0) : \ 0 \leq 0^2 - 4$ is false, so shade the region below the parabola.

Example: Solve the system: $y > x + 2 \quad \text{and} \quad y \leq -x + 4$

Steps:

5. Graph $\ y = x + 2$ with a dashed line and shade above.
6. Graph $\ y = -x + 4$ with a solid line and shade below.
7. The solution is the region where the shaded areas overlap.