Key Concepts of Graphing Linear Inequalities

Why This Matters

Linear inequalities are everywhere in the real world—budgets, speed limits, weight restrictions, and production constraints all involve ranges of acceptable values rather than single answers. When you graph these inequalities, you're visualizing all possible solutions at once, which is exactly what makes them so powerful for modeling constraints. This topic connects directly to systems of equations, linear programming, and optimization problems you'll encounter throughout algebra and beyond.

You're being tested on more than just drawing lines and shading regions. Exam questions probe whether you understand why a boundary is solid versus dashed, how to determine which region satisfies the inequality, and when to apply these techniques to real-world scenarios. Don't just memorize the steps—know what each element of your graph represents and why it matters.

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Understanding Inequality Symbols

The symbols in an inequality tell you two critical things: the direction of the relationship and whether the boundary itself is included. Strict inequalities exclude the boundary; non-strict inequalities include it.

Strict Inequalities: < and >

• Strict inequalities exclude the boundary value—the solution gets infinitely close to the line but never touches it
• Use a dashed line when graphing to visually indicate that points on the line are not solutions
• Common exam trap: students forget that $y < 2x + 1$ means the line itself contains zero solutions

Non-Strict Inequalities: ≤ and ≥

• Non-strict inequalities include the boundary value—points directly on the line are valid solutions
• Use a solid line when graphing to show that boundary points satisfy the inequality
• The "or equal to" portion is what distinguishes these from strict inequalities and changes your line style

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Identifying and Graphing Boundary Lines

Every linear inequality has a boundary line that divides the coordinate plane into two half-planes. The boundary equation is found by replacing the inequality symbol with an equals sign.

Slope-Intercept Form: $y = mx + b$

• The slope $m$ determines the line's steepness—positive slopes rise left to right, negative slopes fall
• The y-intercept $b$ marks where the line crosses the y-axis—plot this point first for easy graphing
• Converting to this form makes graphing straightforward: start at $b$, then use $m$ as rise over run

Vertical Lines: $x = a$

• Vertical lines have undefined slope—they run straight up and down through $x = a$
• These restrict only the x-variable while $y$ can be any value
• Inequalities like $x > 3$ shade everything to the right of the vertical line at $x = 3$

Horizontal Lines: $y = b$

• Horizontal lines have zero slope—they run perfectly flat through $y = b$
• These restrict only the y-variable while $x$ can be any value
• Inequalities like $y \leq -2$ shade everything at or below the horizontal line at $y = -2$

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Determining the Solution Region

Once your boundary line is drawn, you need to identify which half-plane contains the solutions. The test point method is your reliable tool for this.

The Test Point Method

• Choose any point not on the boundary line—the origin $(0, 0)$ is easiest unless the line passes through it
• Substitute the coordinates into the original inequality—if true, shade the side containing your test point
• If false, shade the opposite side—the region not containing your test point holds all solutions

Shading the Correct Region

• The shaded region represents infinitely many solutions—every single point in that area satisfies the inequality
• For $y > mx + b$, shade above the line—these are points where $y$ values exceed the line
• For $y < mx + b$, shade below the line—these are points where $y$ values fall short of the line

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Working with Multiple Inequalities

Real-world problems rarely involve just one constraint. Systems of inequalities require finding where all conditions are satisfied simultaneously.

Compound Inequalities

• "And" compounds require both conditions—the solution is the intersection of individual solution sets
• "Or" compounds require at least one condition—the solution is the union of individual solution sets
• Graphically, "and" creates smaller regions while "or" creates larger, sometimes disconnected regions

Systems of Linear Inequalities

• Graph each inequality on the same coordinate plane—use consistent line styles (solid vs. dashed) for each
• The feasible region is where all shadings overlap—this region satisfies every inequality in the system
• Vertices of the feasible region are often key points for optimization problems

Interpreting Solution Sets

• Every point in the shaded region is a valid solution—you can verify by substituting coordinates into all inequalities
• Solution sets can be bounded or unbounded—bounded regions are enclosed; unbounded extend infinitely
• Context determines meaning—in a budget problem, each point represents a possible combination of purchases

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Quick Reference Table

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Self-Check Questions

1. What two features of a graph change when you switch from $y < 2x + 3$ to $y \geq 2x + 3$?

2. You test the point $(0, 0)$ in an inequality and get a false statement. Which region do you shade, and why?

3. Compare and contrast graphing $x > -2$ versus $y > -2$. How do the boundary lines and shaded regions differ?

4. A system of inequalities produces a triangular feasible region. What must be true about the number of boundary lines, and how do you identify the vertices?

5. If an FRQ asks you to represent "a budget of at most $500 on items costing $20 each (x) and $25 each (y)," write the inequality and explain whether the boundary line should be solid or dashed.