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Systems of Inequalities and Feasible Regions

Swap the equals signs for inequalities and the solution becomes a whole region, not a point. Learn to shade each inequality, find their overlap, and read a real break-even decision off the picture.

From a point to a region

A single linear inequality like y ≥ x + 1 is not satisfied by one point but by half the plane — everything on or above the boundary line. A system of inequalities stacks several such conditions, and its solution set is the overlap of all the shaded halves, called the feasible region.

  1. Graph each boundary line. Use a solid line for ≤ or ≥ (boundary included) and a dashed line for < or > (boundary excluded).
  2. Pick a test point not on the line (often the origin). If it satisfies the inequality, shade its side; otherwise shade the other side.
  3. Repeat for every inequality, shading on the same diagram.
  4. The region covered by all the shadings at once is the feasible region.

Reading a break-even decision

Inequalities are how real constraints look: a budget, a capacity, a minimum order. When cost and revenue meet, you hit the break-even point — found by solving the related system of equations, the boundary case of the inequalities.

A stall: cost C = 40 + 2x, revenue R = 6x (x = items sold).
Profit needs R >= C, i.e.  6x >= 40 + 2x.

Break-even (boundary, R = C):
   6x = 40 + 2x
   4x = 40
   x = 10

So the feasible region for profit is x >= 10.
Selling 10 items breaks even; the 11th item is the first in profit.
Check x = 12:  R = 72,  C = 64,  profit = 8 > 0  ✓
The break-even point is the equality boundary of a profit inequality.