Key Concepts of Graphing Linear Inequalities

Why This Matters

Linear inequalities show up whenever a problem involves a range of acceptable values rather than a single answer. Budgets, speed limits, weight restrictions, and production constraints are all examples. When you graph an inequality, you're visualizing all possible solutions at once, which is what makes them so useful for modeling real-world constraints. This topic connects directly to systems of equations, linear programming, and optimization problems you'll see throughout algebra and beyond.

You're being tested on more than drawing lines and shading regions. Exam questions check 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 word problems. 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 things: the direction of the relationship and whether the boundary value 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 show that points on the line are not solutions.
• Common exam trap: students forget that $y < 2x + 1$ means no point on the line itself is a solution.

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" part 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. You find the boundary equation by replacing the inequality symbol with an equals sign. For example, the boundary line of $y > 2x - 5$ is $y = 2x - 5$.

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

This is the most convenient form for graphing:

1. Identify the y-intercept $b$ and plot that point on the y-axis.
2. Use the slope $m$ (rise over run) to find a second point. A slope of $\frac{3}{2}$ means go up 3 and right 2 from the y-intercept.
3. Connect the points with a solid or dashed line depending on the inequality symbol.

A positive slope rises from left to right; a negative slope falls from left to right.

Vertical Lines: $x = a$

• Vertical lines run straight up and down through the point $x = a$. They have undefined slope.
• These restrict only the x-variable, while $y$ can be any value.
• An inequality like $x > 3$ means you shade everything to the right of the vertical line at $x = 3$.

Horizontal Lines: $y = b$

• Horizontal lines run perfectly flat through the point $y = b$. They have a slope of zero.
• These restrict only the y-variable, while $x$ can be any value.
• An inequality like $y \leq -2$ means you 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 figure out which half-plane contains the solutions. The test point method is your reliable tool for this.

The Test Point Method

1. Pick any point not on the boundary line. The origin $(0, 0)$ is the easiest choice, unless the boundary line passes through it.
2. Substitute that point's coordinates into the original inequality.
3. If the statement is true, shade the side of the line that contains your test point.
4. If the statement is false, shade the opposite side.

Shading the Correct Region

The shaded region represents infinitely many solutions. Every single point in that area satisfies the inequality.

When the inequality is solved for $y$:

• $y > mx + b$ or $y \geq mx + b$: shade above the line (where $y$ values are larger than the line).
• $y < mx + b$ or $y \leq mx + b$: shade below the line (where $y$ values are smaller than the line).

These shortcuts are handy, but the test point method always works, even when the inequality isn't in slope-intercept form.

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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 at the same time.

Compound Inequalities

• "And" compounds require both conditions to be true. The solution is the intersection of the individual solution sets, which creates a smaller region.
• "Or" compounds require at least one condition to be true. The solution is the union of the individual solution sets, which creates a larger (sometimes disconnected) region.

Systems of Linear Inequalities

Here's how to graph a system step by step:

1. Graph each inequality on the same coordinate plane, using the correct line style (solid or dashed) for each one.
2. Shade the solution region for each inequality lightly or with different patterns so you can see where they overlap.
3. Identify the feasible region, which is where all shadings overlap. Only points in this region satisfy every inequality in the system.
4. Find the vertices (corner points) of the feasible region by solving pairs of boundary equations as systems of equations. These vertices matter a lot in optimization problems.

Interpreting Solution Sets

• Every point in the feasible region is a valid solution. You can verify any point by substituting its coordinates into all the inequalities.
• Solution sets can be bounded or unbounded. A bounded region is fully enclosed (like a triangle or rectangle). An unbounded region extends infinitely in at least one direction.
• Context determines meaning. In a budget problem, for instance, each point in the feasible region represents a possible combination of purchases that stays within your constraints.

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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 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. How many boundary lines are needed, and how do you find the vertices?

5. A problem 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.