SAT Practice Questions: Graphing Systems of Inequalities

By Ron Woldoff

The SAT Math exam may ask you to graph a system of inequalities. You solve these in the same way as you would for a system of equations: by graphing each inequality and looking for where the shaded regions intersect.

The following practice questions ask you to find the areas of intersection on the xy-plane, and then to identify which quadrants will contain them.

Practice questions

  1. If the system of inequalities y > x + 3 and y > –x + 2 is graphed in the xy-plane shown here, which quadrants contain all the solutions to the system?
    sat1001-quadrants
    A. Quadrants I and II
    B. Quadrants II and III
    C. Quadrants III and IV
    D. Quadrants I and IV
  2. If the system of inequalities y > x – 5 and y < 2x – 3 is graphed in the xy-plane shown here, which quadrants contain all the solutions to the system?
    sat1001-quadrants
    A.
    Quadrants I, II, and III
    B. Quadrants II, III, and IV
    C. Quadrants I, II, and IV
    D. Quadrants I, III, and IV

Answers and explanations

  1. The correct answer is Choice (A).
    To graph y > x + 3, draw a line going upward and crossing the y-axis at 3; the inequality includes all the solutions above that line. To graph y > –x + 2, draw a line going downward and crossing the y-axis at 2; the inequality includes all the solutions above that line. The result is that all the solutions are contained within a V shape with the vertex at right about (0, 3). This V extends upward into Quadrants I and II.
  2. The correct answer is Choice (B).
    To graph y > x – 5, draw a line going upward and crossing the y-axis at –5; the inequality includes all the solutions above that line. To graph y < –2x – 3, draw a line going downward and crossing the y-axis at –3; the inequality includes all the solutions below that line. The result is that all the solutions are contained within a V shape pointing left, with the vertex slightly right of (0, –4). This vertex is contained within Quadrant IV, and the V extends leftward into Quadrants II and III.