Half-Planes and Systems of Inequalities

In this lesson you will: Graph a linear inequality in two variables as a half-plane with a boundary line; Use a dashed boundary for a strict inequality and a solid one when the boundary is included; Test a point to decide which side to shade, and shade the overlap for a system.

The answer is a region, not a point

Chapter 5 solved inequalities in one variable and shaded an interval on a number line. Two variables need two dimensions, and the same idea produces a shaded piece of the plane.

Start with the boundary: replace the inequality sign by an equals sign and graph that line. It splits the plane into two half-planes. Every point in one of them satisfies the inequality, and no point in the other does.

The line itself belongs to the solution when the sign is <= or >= , and does not when the sign is < or >. Draw it solid in the first case and dashed in the second.

One substitution settles the shading

Do not guess which side to shade, and do not rely on the direction the inequality sign appears to point. Pick any point that is not on the boundary, substitute it, and see whether the statement is true.

If it is true, shade the side containing that point. If it is false, shade the other side. The origin is almost always the easiest test point, and it works unless the boundary passes through it.

For y <= -x+4, test (0,0): is 0 <= 4? Yes, so shade the half-plane containing the origin — the side below the line — and draw the boundary solid.

A system is the overlap

Two inequalities graphed on the same axes give two half-planes. A point solves the system only if it satisfies both, so the solution region is the part of the plane covered twice.

Graph each inequality separately, with its own boundary style and its own shading, and then look for the overlap. The corner of the region sits where the two boundary lines cross, and you find it by solving that system of equations — Lessons 12.2 and 12.3, used as a tool.

For y <= -x+6 and y>x-2, the boundaries meet where -x+6=x-2, so x=4 and y=2. The overlap is the wedge to the left of that corner, below the first line and above the second.

Common mistake — Shading above because the sign says greater than

Asked to graph 2x-y>4, a student reasons that \">\" means \"above\" and shades the half-plane above the line.

The rule of thumb is not invented out of nothing — for an inequality already solved for y, \">\" really does mean above. The mistake is applying it to an inequality that has not been solved for y.

Solve it and watch what happens: -y>-2x+4, and dividing by -1 reverses the sign, giving y<2x-4. The region is below the line, which is the opposite of what the original sign suggested.

The same slip appears whenever the y-term is negative, and it is invisible unless you check.

How to catch it: Test the origin in the inequality as printed. For 2x-y>4, 2(0)-0=0 and 0>4 is false, so the origin is not in the solution region — and the shaded side must be the one that excludes it. One substitution beats any rule about which way the sign points.