Algebra I

Systems of Linear Inequalities

Lesson

A system of inequalities is a collection of inequalities you want to satisfy simultaneously. The solution is the set of all points that satisfy every inequality at once — the feasible region on a graph.

How to graph one

Graph each inequality’s boundary line(replace the inequality with an equals sign).
For $<$ or $>$ use a dashed line. For $\le$ or $\ge$ use a solid line.
Shade the half-plane that satisfies the inequality.
The overlap of all shaded regions is the feasible region.

Worked example — corner point

The feasible region’s corners sit where the boundary lines cross. For $y \ge 2x$ and $y \le x + 3$ , set the boundaries equal:

2x = x + 3 \implies x = 3, \ y = 6

So $(3, 6)$ is a corner.

Testing a point

To check if a point is in the feasible region, substitute it into everyinequality. If it satisfies all of them, it’s a solution. If it fails even one, it’s out.

The corner pointsmatter because in real-world optimization (linear programming), the best answer always lives at a corner. That’s why this topic is the bridge from algebra to applications.

How to type your answer

Enter a point as x,y — for example, 3,6.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{Boundary intersection of } y = x \text{ and } y = 4

Problem 2

\text{Boundary intersection of } y = 2x \text{ and } y = 6

Problem 3

\text{Boundary intersection of } y = x + 1 \text{ and } y = 5

Problem 4

\text{Boundary intersection of } y = -x \text{ and } y = -2

Practice

Standard problems matching the lesson.

Problem 5

Corner of y ≥ x+1 and y ≤ 5

Problem 6

Corner of y ≥ 2x and y ≤ x + 3

Problem 7

Corner of y ≥ x and y ≤ -x + 8

Problem 8

Corner of y ≤ 3x and y ≥ x + 4

Problem 9

Top-right corner of y≥0, x≥0, y≤4, x≤5

Problem 10

\text{Intersection of } x = 3 \text{ and } y = 6

Problem 11

Corner of y ≥ -x + 4 and y ≤ x

Problem 12

Corner of y ≥ x - 3 and y ≤ -2x + 9

Problem 13

Corner of 2x+y≤10, x+y≤6

Problem 14

Intersection of x+y=7 and x-y=1

Problem 15

Top-right corner of y≥0, y≤x, x≤4

Problem 16

Corner of y ≤ x + 2 and y ≥ -x + 4

Problem 17

2x + 3y ≤ 12. If x = 3, max-y corner (x,y)?

Problem 18

Max-x corner of x≥0, y≥0, x+y≤10

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Intersection of 3x+y=12 and x+2y=9

Problem 20

Corner of x+y≤8 and 2x+y≤10 (both tight)

Problem 21

Corner of y ≥ 2x - 1 and y ≤ x + 3

Problem 22

Intersection of x+2y=10 and 3x+y=15

Problem 23

x + 2y ≤ 10. If x = 2, max-y corner (x,y)?

Problem 24

Intersection of y = -2x + 10 and y = x + 1

Problem 25

Top-right corner of y≥0, y≤x, x≤5

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