Systems of Inequalities — SAT Math Explained
Two or more linear inequalities considered simultaneously. The solution is the overlapping region (intersection) of all individual shaded regions — every point that satisfies ALL inequalities at once.
The Core Idea
Each inequality restricts the coordinate plane. When you have multiple restrictions simultaneously, only the region satisfying all of them qualifies. This overlap is the feasibility region.
Step-by-Step: How to Approach Systems of Inequalities
Graph the first inequality: draw the boundary line (solid/dashed), shade the solution region
On the SAME coordinate plane, graph the second inequality the same way
Identify the region where the shading overlaps — this is the solution set
To confirm, pick a point from the overlapping region and verify it satisfies ALL inequalities
No Solution Case
If the shaded regions don't overlap at all, the system has no solution — no point satisfies all conditions simultaneously
Real World Application
Linear programming: maximizing profit or minimizing cost subject to constraints (budget, capacity, time)
Scheduling: finding times that satisfy all participants' availability
Nutrition planning: finding meal combinations that meet all dietary requirements
Common Errors to Avoid
Only shading for one inequality and forgetting the other
Identifying the union (all shaded area) instead of the intersection (overlap only)
Not checking a point from the final region in all original inequalities
Practice: Systems of Inequalities
5 SAT-style questions. Select your answer and get an instant explanation.
Which point satisfies BOTH inequalities: y > x and y < 5?
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