Graphical Solution of Linear Inequalities in Two Variables

A linear inequality in two variables, such as $2x + y \le 10$, is satisfied not by isolated points on a line but by an entire region of the Cartesian plane — because for almost every value of $x$, a whole range of $y$-values makes the inequality true. Solving such an inequality graphically means identifying and shading exactly that region.

Step 1 — Draw the boundary line. Replace the inequality sign with an equals sign to get the corresponding linear equation — e.g. $2x+y \le 10$ has boundary equation $2x+y=10$ — and plot this straight line using its intercepts or a table of values.

• If the inequality is slack ($\le$ or $\ge$), draw the line solid, since points on the line satisfy it and belong to the solution.
• If the inequality is strict ($<$ or $>$), draw the line dashed (broken), since points on the line do not satisfy it and must be visibly excluded.

Step 2 — Test a convenient point. The boundary line divides the plane into exactly two half-planes. To find which one is the solution, pick any point not on the line — the origin $(0,0)$ whenever the line does not pass through it, since substituting $x=0,y=0$ is the fastest check — and substitute into the original inequality.

• If the test point satisfies the inequality, shade the half-plane containing that point.
• If it fails, shade the other half-plane.

Worked illustration. Solve $2x + y \le 10$ graphically. Boundary line $2x+y=10$ passes through $(5,0)$ and $(0,10)$; since the inequality is slack, draw it solid. Test the origin: $2(0)+0 = 0 \le 10$ — true. So shade the half-plane containing the origin.

Systems of two-variable inequalities. When two or more linear inequalities in $x,y$ must hold simultaneously, each is graphed by Steps 1–2 on the same axes, and the overall solution is the region where all the individual shaded regions overlap — the common (intersection) region. If this common region is enclosed on every side, it is a bounded polygon, and its corner points — found by solving pairs of boundary equations simultaneously — matter greatly in later applications such as linear programming. …