Table, Algebra, and Graph of an Inequality

A table of pairs and a shaded region carry the same information because both are just ways of listing every point that makes the inequality true. The table shows a sample of those points. The graph shows all of them at once: the boundary line marks where the two sides of the inequality are equal, and the shading marks every point on the true side. Pick any point from the table and it will land inside the shaded region, and pick any point from the shaded region and it will satisfy the inequality.

To recover the inequality from a graph, read the boundary line first: find its slope and y-intercept to write the equation y = mx + b. Then decide the direction. Test one point that is clearly not on the line, often (0, 0) if the line does not pass through it. If that point satisfies the relation once you swap the equals sign for <, ≤, >, or ≥, shade its side; if not, shade the other side. Solid lines mean the boundary points themselves count as solutions, so the inequality uses ≤ or ≥. Dashed lines mean the boundary is excluded, so it uses strict < or >.

This connects directly to systems. When two inequalities each shade a region, their overlap is the full solution set of the system. The corner of that overlap sits exactly where the two boundary lines cross, which is the same point you get by solving the matching system of equations. So solving a system algebraically and reading a shaded overlap graphically are two routes to the same answer.