Solving Systems of Inequalities

A system of inequalities is a set of two or more inequalities that must all be true at the same time. Unlike a system of equations — which usually has a single point as its solution — a system of inequalities has infinitely many solutions. The solution is an entire region of the graph where all the inequalities are satisfied simultaneously.

The most practical way to find that region is by graphing.

Quick Symbol Review

Before graphing, make sure you're comfortable with the four inequality symbols:

  • \(<\) — less than
  • \(>\) — greater than
  • \(\leq\) — less than or equal to
  • \(\geq\) — greater than or equal to

The difference between \(<\) and \(\leq\) matters when graphing: a \(\leq\) or \(\geq\) inequality includes the boundary line itself (draw it solid), while \(<\) or \(>\) does not (draw it dashed).

Graphing a Single Inequality

The process for graphing one inequality:

Step 1 — Rewrite in slope-intercept form (\(y = mx + b\)) if it isn't already.

For example, \(x + y \leq 10\) becomes \(y \leq -x + 10\).

Step 2 — Graph the boundary line by treating the inequality as an equation: \(y = -x + 10\). Draw the line solid if the symbol is \(\leq\) or \(\geq\), dashed if it's \(<\) or \(>\).

Step 3 — Shade the correct side. Since the inequality says \(y \leq\) something, shade below the line (where y values are smaller). If it said \(y \geq\), you'd shade above.

Step 4 — Verify with a test point. Pick any point not on the line — \((0, 0)\) is usually easiest — and plug it into the original inequality. If it makes the inequality true, you shaded the right side. If not, shade the other side.

For \(y \leq -x + 10\): plug in \((0,0)\) to get \(0 \leq 10\). True — so the shading below the line is correct.

Graph of y ≤ -x + 10, shaded below the line

A Second Example

Find all values satisfying \(y \geq \frac{-3}{2}x + 6\).

This is already in slope-intercept form. Graph the line \(y = \frac{-3}{2}x + 6\) as a solid line (because of \(\geq\)), then shade above it since y must be greater than or equal to the line.

To verify, test the point \((5, 3)\):

$$3 \geq \frac{-3}{2}(5) + 6 = -1.5$$

\(3 \geq -1.5\) is true, and \((5,3)\) is above the line — the shading is correct.

Graph of y ≥ (-3/2)x + 6, shaded above the line

Combining Into a System

When you have two inequalities, graph both on the same set of axes. Each one shades a region, and the solution to the system is wherever both shaded regions overlap.

Taking the two examples above together:

Graph showing the overlapping solution region for both inequalities

The darker overlapping area is the solution set — every point in that region satisfies both inequalities at the same time. To confirm any particular point is a solution, plug it into both inequalities and check that both are true.

A few things to keep in mind as you work through these:

  • If the shaded regions don't overlap at all, the system has no solution.
  • If one region is entirely inside the other, the solution set is the smaller region.
  • Always check your shading with a test point — it's easy to shade the wrong side, especially when the inequality isn't in slope-intercept form yet.

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