A clear, step-by-step walkthrough of solving systems of linear inequalities by graphing, with a fully worked example and the most common mistakes students make.
A system of inequalities is two or more inequalities that share the same variables. Instead of looking for a single point that satisfies both (like in a system of equations), you're looking for a whole region of the coordinate plane where all the inequalities are true at the same time.
That region is called the solution set, and it shows up on a graph as an overlapping shaded area.
Every system of inequalities problem follows the same four steps. Once you know the process, it becomes automatic.
Graph the boundary line for each inequality. Treat the inequality like an equation and graph the line. Use a solid line for \(\leq\) or \(\geq\) (the boundary is included). Use a dashed line for \(<\) or \(>\) (the boundary is not included).
Shade the correct half-plane for each inequality. Pick a test point not on the line (the origin, \((0, 0)\), works great as long as it's not on the line). Plug it into the inequality. If the test point makes it true, shade the side containing that point. If it makes it false, shade the other side.
Identify the overlapping region. The solution to the system is the area where both shaded regions overlap. Every point in that overlap satisfies both inequalities simultaneously.
Check your answer. Pick any point from the overlapping region and plug it into both inequalities. Both should be true. If one fails, recheck your shading.
The boundary lines in your system will usually cross each other. That intersection point is the corner of your solution region, and you may need to find it exactly.
To find it, set the two equations equal to each other and solve:
From the example above: \(2x + 1 = -x + 4\)
\(3x = 3\) → \(x = 1\), then \(y = 2(1) + 1 = 3\)
The lines intersect at \((1, 3)\). Note that this corner point belongs to the solid boundary (line 1) but NOT the dashed boundary (line 2), so whether it's included in the solution depends on which line it sits on.
A common way to remember it: if there's an "equals" in the symbol, the line is solid. If not, it's dashed.
Shading the wrong side. Always test a point. Don't just assume "greater than means above the line" — that's only true when the inequality is already solved for \(y\). If your equation is in standard form like \(2x - y > 4\), you need to test a point to be sure.
Forgetting to flip the inequality sign. When you multiply or divide both sides by a negative number while solving for \(y\), the inequality sign reverses. Missing this step leads to shading the wrong side entirely.
Using a solid line when you need a dashed one (or vice versa). Read the symbol carefully before you graph. Mixing these up is one of the most common point-losses on tests.
Not checking the overlap. Students sometimes shade both regions and forget that the answer is only where they overlap. Identify and clearly mark that region.
If the shaded regions don't overlap at all, the system has no solution. This can happen when the boundary lines are parallel and the inequalities point in opposite directions. For example, \(y > x + 5\) and \(y < x - 3\) — those regions never meet.
If the shaded regions together cover the entire plane, every point is a solution. This is less common, but possible — for example, \(y \geq x\) and \(y \leq x + 10\) together include an infinitely wide band.
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