ACT Math: Solve Systems of Inequalities by Finding the Overlapping Region

A system of inequalities has multiple inequality constraints. The solution is the region where all constraints are satisfied simultaneously. Example: x>0, y>0, x+y<5. Graph each: x>0 is right of the y-axis, y>0 is above the x-axis, x+y<5 is below the line x+y=5. The overlapping region (where all three are true) is the solution. The key skill is identifying the correct region by shading and finding the overlap.

Process: (1) Convert each inequality to an equation and graph the boundary line (solid for ≤ or ≥, dashed for < or >). (2) Test a point to determine which side of the line satisfies the inequality. (3) Shade that side. (4) The solution is the region shaded by all inequalities.

Step 1: Graph the boundary line for each inequality. Use a solid line for ≤/≥ and a dashed line for x and y<2. Step 1: Graph y=x (dashed) and y=2 (dashed). Step 2: Test (0,1): Is 1>0? Yes, so shade above the line y=x. Is 1<2? Yes, so shade below y=2. Step 4: The region above y=x and below y=2 is the solution. This method works for every system of inequalities on ACT Math.

Practice: Solve x≥0, y≥0, x+y≤4. Graph: x=0 (solid), y=0 (solid), x+y=4 (solid). Test (1,1): 1≥0? Yes. 1≥0? Yes. 1+1≤4? Yes. Shade the region where all three are true (the triangle with vertices at (0,0), (4,0), (0,4)).

Mistake 1: Shading the wrong side of a boundary line (testing the wrong region). Fix: Always test a point to confirm which side satisfies the inequality. Mistake 2: Using a solid line for < or > (should be dashed). Fix: < and > are dashed; ≤ and ≥ are solid. Mistake 3: Finding the wrong overlap (shading multiple regions instead of the one region that satisfies all inequalities). Fix: Use different colors or patterns for each inequality to see the overlap clearly. Mistake 4: Forgetting to check if a vertex point (corner of the region) is included (solid vs. dashed boundary). These four mistakes cause 90% of system-of-inequalities errors.

Drill: Graph y>2x, y<-x+4, x>-1. Step 1: Graph y=2x (dashed), y=-x+4 (dashed), x=-1 (dashed). Step 2: Test (0,0): 0>0? No, shade below y=2x. 0<4? Yes, shade below y=-x+4. 0>-1? Yes, shade right of x=-1. Step 4: Identify the overlapping region.

Systems of inequalities appear on most ACT Math sections, usually in the medium-difficulty range. Students who own the four-step method solve these quickly; students who guess or struggle waste time. Each inequality system you solve correctly is a point that tests a learnable, mechanical skill, not conceptual understanding.

Spend one week graphing ten systems using the four-step method. By test day, you'll graph systems so quickly that you'll answer these questions in under 2 minutes, freeing up time for harder problems. That speed advantage will show in your overall score.