Graphical Inequalities

Inequalities are not equations. They portray relationships in which one side may be greater than, less than, or possibly equal to the other. A typical inequality symbol might be > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to).
You can represent inequalities on number lines, where a hollow circle indicates ‘greater than’ or ‘less than’ and a solid circle indicates ‘greater than or equal to’ or ‘less than or equal to’.
Graphical inequalities are depicted in two dimensions, using x and y axes. The area that satisfies the inequality is often shaded.
When sketching a graphical inequality, first draw the line. If the inequality is ‘greater than’ or ‘less than’, the line should be dashed. If it’s ‘greater than or equal to’ or ‘less than or equal to’, the line should be solid.
To determine which area to shade, choose a test point not lying on the line (often the origin, (0,0), is chosen). Substitute the coordinates of this point into the inequality. If the inequality is true, shade the region containing the test point. If false, shade the other region.
Intersection of inequalities can be drawn on the same set of axes. The area of intersection is the region that satisfies all inequalities simultaneously.
You can also represent quadratic inequalities graphically. First, sketch the graph of the quadratic equation (if the inequality symbol were replaced by an equals symbol). Whether the curve is dashed or solid depends on whether the inequality sign includes equality or not.
When shading for a quadratic inequality, you generally divide the graph into three regions: above the graph, on the graph, and below the graph. The decision about which region(s) to shade depends on whether the inequality symbol is ‘<’, ‘>’, ‘≤’, or ‘≥’ and on whether the curve opens upward or downward.
To solve inequalities, you work mainly with the equation part (as if the inequality sign were an equals sign), then consider the inequality part at the end of the solution process.
Always pay attention to the specific rules of inequality operations when solving (like, if you multiply or divide by a negative number, you must reverse the inequality sign).