What is a system of inequalities?

A system of inequalities is two or more inequalities with the same variables considered together.

• Goal: Find all the values of variables that satisfy all inequalities at the same time.

• Usually involves x and y.

• Solutions form a region on the coordinate plane, not just a single point.

Example:

• The solution is the overlapping region where both inequalities are true.

Why systems of inequalities matter

• Represent real-world constraints:

  • Budget limitations

  • Speed or production constraints

  • Area or distance requirements

• Helps visualize feasible regions in algebra and applied math.

How to solve a system of inequalities

Step 1: Graph each inequality separately

1. Rewrite each inequality in slope-intercept form: y = mx + b.

2. Graph the boundary line:

   • < or > → dashed line

   • ≤ or ≥ → solid line

3. Shade the region that satisfies the inequality:

   • Above the line if y > mx + b or y ≥ mx + b

   • Below the line if y < mx + b or y ≤ mx + b

Step 2: Identify the solution region

• The solution of the system is the overlapping shaded region from all inequalities.

• Every point in this region satisfies all inequalities at the same time.

Example 1: Solve and graph

Step 1: Graph each boundary line

• y = x + 1 → solid line, shade above

• y = −2x + 5 → dashed line, shade below

Step 2: Identify overlap

• The solution region is where the shading overlaps.

✅ This shows all points (x, y) that satisfy both inequalities.

Example 2: Solve a real-world system

A company produces two types of products, A and B:

Step 1: Rewrite as equations for boundaries:

• 2A + B = 20 → boundary line

• A + 2B = 18 → boundary line

Step 2: Graph inequalities:

• Shade below each line (because of ≤)

• Solution region = overlapping area

✅ This represents all combinations of products A and B that satisfy both constraints.

Tips for graphing systems of inequalities

1. Always draw the boundary line first

2. Use dashed for < or > and solid for ≤ or ≥

3. Test a point (like (0,0)) to check which side to shade

4. Look for the overlapping region — that’s the solution

Common beginner mistakes

1. ❌ Forgetting to shade the correct side of the line

2. ❌ Not using dashed/solid lines correctly

3. ❌ Confusing the solution of one inequality with the system

4. ❌ Forgetting that the solution is a region, not a single point

Why systems of inequalities matter

• Solve problems with multiple restrictions simultaneously

• Foundation for linear programming and optimization problems

• Visualizes feasible solutions clearly