Matching Inequalities to Number-Line Graphs

A number-line graph is just a picture of every number that makes an inequality true. The boundary point is the number x is being compared to. The circle there tells you whether that exact number is a solution: filled means yes, it's included, which happens with ≤ or ≥. Hollow means no, it's excluded, which happens with < or >. Everything else about the graph is decided by testing numbers, not by guessing from the symbol's shape.

To get the shading direction right, pick a number on one side of the boundary and check whether it satisfies the inequality. If x ≥ -5/2, try 0: is 0 ≥ -5/2? Yes. So shading goes right, toward the larger numbers, and the circle at -5/2 is filled because -5/2 itself also satisfies the inequality. If instead you had x < -5/2, try 0 again: 0 < -5/2 is false, so shading goes left, and the circle stays hollow since -5/2 is not less than itself.

This same read-the-inequality-literally habit carries over to solving and graphing compound inequalities and absolute value inequalities later in this unit, where you'll often get two boundary points instead of one. Get single-inequality graphs solid first, because every later version still comes down to the same two questions: does the boundary count, and which direction actually satisfies the statement.