Systems of Linear Inequalities

Say you have to spend on pens at each and notebooks at each, and you need at least three notebooks. There isn’t one right answer. Five pens and four notebooks works, so does none and six, and so do plenty of others. When the conditions are inequalities instead of equations, the solution is a whole region of points. A system of inequalities describes that region, and graphing is the natural way to find it. This lesson builds on linear inequalities in one variable and on graphing systems of equations.

What Is a System of Inequalities?

A system of linear inequalities is two or more linear inequalities in the same variables, for example:

A solution is any point that makes every inequality true. There are usually infinitely many, and together they fill a region of the plane called the solution region (or, in applications, the feasible region).

Graphing One Inequality in Two Variables

Each inequality on its own shades half of the plane, called a half-plane. Two decisions turn the equation of a line into that picture: how to draw the line, and which side to shade.

If the inequality isn’t solved for , use a test point. Pick any point not on the line, usually , and substitute it. If the inequality comes out true, shade the side containing the test point; if false, shade the other side.

How to Solve a System of Inequalities by Graphing

1. Graph the boundary line of the first inequality: solid for or , dashed for or .
2. Shade the side that makes the first inequality true.
3. Repeat for every other inequality, on the same axes and in a different color.
4. Find the overlap. The region shaded by every inequality is the solution of the system.
5. Check a point from the overlap in every inequality.

Example 1: Two Inequalities in Slope-Intercept Form

Sketch the solution to the system and .

Step 1 — first inequality: draw as a solid line (the symbol is ) and shade below it.

Step 2 — second inequality: draw as a dashed line (the symbol is ) and shade above it.

Step 3 — the overlap is the wedge to the right of where the lines cross, which is at . That corner point itself is not a solution, because it lies on the dashed line.

Step 4 — check a point in the overlap, say :

Which Point Is a Solution to the System of Inequalities?

You don’t need a graph to test a point. Substitute it into every inequality, and it’s a solution only if all of them are true.

The last two rows show why the line style matters. A point on a solid boundary counts; a point on a dashed boundary doesn’t.

Example 2: A Closed Region

Graph the system , and , and find the corners of the solution region.

Step 1 — : the boundary passes through and . The test point gives , which is true, so shade toward the origin.

Step 2 — : a solid vertical line at , shaded to the right.

Step 3 — : a solid horizontal line at , shaded above.

Step 4 — the overlap is a triangle. Its corners, the vertices, are where the boundary lines meet: , and .

Bounded regions like this one are the starting point of linear programming, where you look for the corner that maximizes profit or minimizes cost.

Which System of Inequalities Is Shown by the Graph?

Many questions give you the shaded graph and ask for the system. Work one boundary at a time.

Step 1 — the blue line passes through and , so its equation is . It’s dashed and shaded below, so the inequality is .

Step 2 — the green line passes through and : slope , intercept . It’s solid and shaded above, so the inequality is .

Step 3 — the system:

Step 4 — confirm with a point in the overlap, such as : ✓ and ✓.

For multiple-choice versions (“which of the following systems of inequalities would produce the graph?”), testing one point from the overlap and one point outside it in each answer choice usually eliminates the wrong options fast.

When a System of Inequalities Has No Solution

Solve and . Both boundaries have slope , so they’re parallel. The first shades above the upper line and the second shades below the lower line. The two regions never meet, so the system has no solution. Compare that with and , where the shading overlaps in the strip between the parallel lines, giving infinitely many solutions.

Worked Example: Staying Within a Budget

You have to spend. Pens cost , notebooks cost , and you need at least notebooks. Write a system of inequalities and decide whether 5 pens and 4 notebooks, or 8 pens and 3 notebooks, is possible.

Step 1 — variables: pens and notebooks.

Step 2 — one inequality per condition:

Step 3 — test 5 pens and 4 notebooks:

Step 4 — test 8 pens and 3 notebooks:

The first plan fits the budget exactly. The second costs , one dollar too much.

Common Mistakes to Avoid

• Using the wrong line style. and need dashed lines; and need solid ones.
• Shading by the symbol when the inequality isn’t solved for . In , “less than” does not mean shade below. Solve for first (the sign flips: ) or use a test point.
• Forgetting to flip the inequality when multiplying or dividing by a negative number.
• Using a test point on the boundary. can’t be the test point when the line passes through the origin. Use or instead.
• Reporting one point as “the” answer. The solution is the whole overlapping region, unless the question asks for something specific such as a vertex.

Practice Problems

Work each one before opening the answer.

Problem 1. Is a solution of the system and ?

Problem 2. Is a solution of the system and ?

Problem 3. Describe how to graph .

Problem 4. Which of the points , and are solutions of and ?

Problem 5. A region lies above the dashed line and below the solid line . Write the system of inequalities.

Problem 6. Does the system and have any solutions?

Systems of inequalities describe what’s allowed; systems of equations pin down exactly what is. Real problems mix both, and turning a word problem into the right system is the skill that ties this whole chapter together.