Solving Systems of Inequalities- Graphical and Algebraic Methods

What a System of Inequalities Actually Is

A system of inequalities is just two or more inequalities that you need to satisfy at the same time. It's not complicated. You already know how to graph lines. Now you're going to shade regions instead.

Think of it like this: every point in the shaded area is a valid solution. Every point outside is not. That's it. Nothing mystical about it.

The Two Methods You Need to Know

There are exactly two ways to solve these problems:

• Graphical method — plot it, shade it, done
• Algebraic method — test points, no graphing required

Most textbooks push the graphical method because it's visual. But you'll encounter situations where graphing is impractical. You'll need both.

Graphical Method: Step by Step

This is the method teachers love because you can actually see the solution region. Here's how it works.

Step 1: Convert to Slope-Intercept Form

Take each inequality and rewrite it as y = mx + b. This tells you the boundary line.

Example: 2x + 3y ≤ 12 becomes 3y ≤ -2x + 12, then y ≤ (-2/3)x + 4

Step 2: Graph the Boundary Line

Use a solid line for ≤ or ≥ (the boundary is included). Use a dashed line for < or > (the boundary is not included).

This distinction matters. Lose points on tests for getting this wrong.

Step 3: Test a Point to Determine Shading

Pick any point not on the line. The origin (0,0) works most of the time unless the line passes through it.

Plug it into the inequality. If it's true, shade that side. If it's false, shade the other side.

Step 4: Find the Overlapping Region

Once you've shaded all inequalities, the solution is where all shadings overlap. That region contains every point that satisfies every inequality simultaneously.

Algebraic Method: Testing Without Drawing

Sometimes you don't have graph paper. Sometimes the numbers are ugly. Sometimes you just need to verify a specific point.

How It Works

You take a potential solution (x, y) and plug it into every inequality in the system. If it satisfies all of them, it's part of the solution set. If it fails even one, it's not.

That's the entire method. No shading, no coordinate plane.

Example

System: x + y > 3 and 2x - y ≤ 1

Test point (2, 2):

• 2 + 2 > 3 ✓
• 2(2) - 2 ≤ 1 → 4 - 2 ≤ 1 → 2 ≤ 1 ✗

Point (2, 2) fails. It's not a solution.

Test point (3, 1):

• 3 + 1 > 3 ✓
• 2(3) - 1 ≤ 1 → 6 - 1 ≤ 1 → 5 ≤ 1 ✗

Still fails. Try (1, 3):

• 1 + 3 > 3 ✓
• 2(1) - 3 ≤ 1 → 2 - 3 ≤ 1 → -1 ≤ 1 ✓

Point (1, 3) works. That's your answer.

Graphical vs Algebraic: When to Use What

Common Mistakes That Cost You Points

• Using the wrong line type — solid vs dashed. This is the most common error. Solid lines mean the line is included in the solution.
• Shading the wrong side — always test a point. Never guess based on the slope.
• Forgetting to flip the inequality — when you multiply or divide by a negative number, the sign reverses. Every teacher watches for this mistake.
• Missing the overlap region — each inequality gets its own shading. The solution is where they all intersect.

Getting Started: A Worked Example

Let's solve this system graphically:

y ≥ x - 2
y < -x + 4

First inequality: y ≥ x - 2

• Graph y = x - 2 as a solid line (≥ includes the boundary)
• Test (0, 0): 0 ≥ 0 - 2 → 0 ≥ -2 ✓
• Shade above the line

Second inequality: y < -x + 4

• Graph y = -x + 4 as a dashed line (< excludes the boundary)
• Test (0, 0): 0 < -0 + 4 → 0 < 4 ✓
• Shade below the line

The solution region is where the shading from both inequalities overlaps. You'll see a wedge-shaped area bounded by the two lines.

Any point in that region works. (0, 1) works. (1, 2) works. (2, 1) works. Pick any point and verify with the algebraic method if you want to double-check.

When the Solution Region Is Empty

Sometimes no point satisfies all inequalities at once. This happens when the constraints contradict each other.

Example: y > 5 and y < 3

Nothing is simultaneously greater than 5 and less than 3. When you graph this, the shaded regions will never overlap. The system has no solution.

You'll see this in linear programming problems. The feasible region disappears entirely.

Real-World Applications

You won't graph inequalities for fun. Here's where this actually shows up:

• Budget constraints — spending on X plus spending on Y must stay under a limit
• Resource allocation — production of A and production of B must meet minimum demand
• Time management — hours on task A plus hours on task B must fit within available time

The graphical method shows you the feasible region at a glance. Business problems often have two variables for exactly this reason.

Quick Reference

• ≤ and ≥ use solid lines
• < and > use dashed lines
• Always test a point to confirm shading direction
• The solution is the overlap, not one shading or the other
• Algebraic method = plug in and check
• Graphical method = plot, shade, identify overlap

Master both methods. Know when each one applies. The test won't always let you use a graph, and sometimes you need to see the region to answer the question.