Question

What are systems of inequalities?

Answer 1

A system of inequalities is a set of two or more inequalities that are considered at the same time. A solution to the system is any value (or set of values) that satisfies every inequality in the system simultaneously.

Key ideas
- Each inequality defines a region of possible values. The solution to the system is the intersection of those regions.
- For one variable, each inequality gives an interval; intersect the intervals to get the solution set.
- For two variables (usually x and y), each inequality defines a half‑plane (or region bounded by a curve); graph the boundaries and shade the appropriate sides. The solution region is where all shaded regions overlap.
- Boundaries: use a solid line for ≥ or ≤ (boundary included) and a dashed line for > or < (boundary excluded).

Simple examples
1) One variable
System: x − 3 > 2 and 2x + 1 ≤ 9
Solve each:
x − 3 > 2 → x > 5
2x + 1 ≤ 9 → x ≤ 4
Intersection: no x satisfies both → no solution.

2) Two variables (graphical)
System: y > 2x − 1 and y ≤ −x + 4
- Graph the lines y = 2x − 1 (dashed) and y = −x + 4 (solid).
- Shade above the first line and at or below the second line.
- The solution is the region where the two shaded areas overlap (points between the two lines satisfying both inequalities).

Other notes
- Systems of linear inequalities in two or more variables produce convex polygonal regions (polyhedra in higher dimensions). These are central to linear programming (finding maxima/minima on feasible regions).
- A system may have no solution (empty), a bounded solution region, or an unbounded solution region.

If you want, I can show a step‑by‑step graph of the two‑variable example or solve a specific system you give.