SAT Algebra — inequalities in two variables: graphs and shaded regions

Linear inequalities in one or two variables are one of the five skill areas of the SAT Algebra domain, and the two-variable case is often shown as a picture: a line in the xy-plane with one side shaded. This material trains the reading of such graphs in both directions, from the picture to the inequality and from the inequality to the region.

Four questions come with a coordinate grid. You find the equation of a boundary line from two marked points, decide the direction of the sign from the side that is shaded, and decide whether it is strict from the line style: a dashed line leaves the boundary out, a solid line keeps it. One graph uses a line written in standard form, where the test point (0, 0) settles the direction better than the words "over" or "under"; another shows a system of two inequalities with one solid and one dashed boundary; a third uses a vertical and a horizontal boundary and asks which point lies in the shaded region.

The other questions work without a picture: testing points against a strict inequality when one of them sits exactly on the boundary, rewriting an inequality in standard form for y (and reversing the sign when dividing by a negative number), finding the values of a parameter that keep a point inside a system, recognizing the inequality whose graph has a dashed horizontal boundary, and spotting a system with parallel boundaries that has no solution at all. A context about a bakery with a limit on oven time and a minimum number of items is translated into a system, then used to find the greatest possible number of one item and to interpret a point of the solution region.

The quiz has twelve multiple-choice questions with worked explanations, each distractor built from a typical slip: the wrong side of the line, the wrong line style, a forgotten sign reversal, "at most" read as "at least". The flashcards collect the rules for boundaries, shading, test points and systems. A calculator is allowed throughout the SAT Math section, but these questions reward reading the graph correctly more than computing.

The material is based on the College Board's published description of the SAT Math section (Algebra domain: linear inequalities in one or two variables). It is independent practice and is not produced or endorsed by the College Board.

• Write the inequality represented by a shaded half-plane, choosing the sign from the shaded side and its strictness from a dashed or solid boundary.
• Identify the system of two inequalities whose solution set is a shaded region, and the points that belong to it.
• Use a test point to decide the direction of an inequality written in standard form.
• Rewrite an inequality for y, reversing the sign when dividing by a negative number.
• Translate a context into a system of inequalities and use it to find a maximum and to interpret a solution.

Practice material written by Zestly, based on the College Board's description of the digital SAT Math section, Algebra domain (skill: linear inequalities in one or two variables).