Inequality graphing involves plotting regions of the coordinate plane that satisfy an inequality or a system of inequalities. Here's how to graph an inequality:

Steps to Graph an Inequality:

1. Convert to Equality: Rewrite the inequality as if it were an equality (e.g., $y \leq 2x + 3$ becomes $y = 2x + 3$).
2. Plot the Boundary Line:
   • Use the equation to graph the boundary line.
   • Solid Line: Use for $\leq$ or $\geq$.
   • Dashed Line: Use for $<$ or $>$.
3. Shade the Correct Region:
   • Choose a test point (like $(0, 0)$, unless it lies on the line) and substitute it into the inequality.
   • If the point satisfies the inequality, shade the region containing the point. Otherwise, shade the opposite region.
4. Combine Regions (for Systems of Inequalities): For multiple inequalities, plot each one separately, then find the overlapping shaded region.

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Example 1: Graph $y > 2x + 1$

1. Rewrite as equality: $y = 2x + 1$.
2. Plot a dashed line for $y = 2x + 1$ (since $>$ is strict).
3. Test point $(0, 0)$:
   • Substitute: $0 > 2(0) + 1$ → $0 > 1$, which is false.
   • Shade the region above the line.

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Example 2: Graph $x + y \leq 4$ and $y \geq 1$

1. Graph $x + y = 4$:
   • Solid line since $\leq$.
   • Test point $(0, 0)$: $0 + 0 \leq 4$ is true, so shade below.
2. Graph $y = 1$:
   • Solid horizontal line at $y = 1$.
   • Shade above the line.
3. The solution region is the overlap.

Would you like me to solve and plot a specific inequality or system of inequalities for you?

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Related Questions

1. How do you graph absolute value inequalities like $|x - 2| < 3$?
2. What’s the difference between graphing $y > x^2$ and $y \geq x^2$?
3. How do you solve and graph a system of linear inequalities?
4. How do you determine whether a point is in the solution region of an inequality?
5. Can you graph nonlinear inequalities, such as $x^2 + y^2 \leq 9$?

Tip: Always check the boundary line and shading by substituting a test point into the inequality!