A system of linear inequalities is a set of two or more inequalities that must be true at the same time. Each inequality represents a half-plane on a coordinate graph, separated by a boundary line. The solution to the system is the region where all the shaded half-planes overlap.
This matters because many real situations have several limits at once, such as budget, time, space, or resource constraints.
To graph a linear inequality, first graph its boundary line, then decide which side of the line satisfies the inequality. A solid boundary line means points on the line are included, while a dashed boundary line means they are not included. The feasible region is the overlapping shaded region that satisfies every inequality in the system.
Testing a point, often (0, 0) when it is not on a boundary line, helps confirm which side should be shaded.
Understanding Math: Systems of Linear Inequalities
A graph is useful because it turns several written restrictions into a picture of possible choices. Every point represents one pair of values. The horizontal coordinate might represent hours spent studying, while the vertical coordinate represents hours spent working.
A point belongs in the final region only when its coordinates pass each rule. This gives a direct check for any proposed answer. Substitute the horizontal value and vertical value into each statement, then decide whether every comparison is true.
One failed comparison removes that point, even if it appears to fit the other restrictions. This is why the overlap must be treated as one complete set of conditions rather than as separate shaded areas.
Not every boundary line is easiest to graph using slope and intercept form. A rule involving only the horizontal variable creates a vertical boundary. For example, a condition saying the horizontal value is no more than four has a line through four on the horizontal axis.
A rule involving only the vertical variable creates a horizontal boundary. When rearranging an inequality to isolate a variable, take special care when multiplying or dividing by a negative number. The direction of the inequality must reverse.
Missing this change sends the shading to the wrong side and can change the entire answer. It helps to write the equality version first, graph that line carefully, then use a simple test point away from the line.
The edges where boundary lines meet are especially important. These meeting points are called vertices or corner points of the feasible region. They often show the most extreme possible values in a problem.
For example, a business may want the greatest profit while staying within limits on materials and labor. In a linear optimization problem, the best value occurs at a vertex when a best value exists. Find candidate vertices by solving pairs of boundary equations, then check each candidate against every original inequality.
Some systems have no feasible points because their restrictions conflict. Others produce a region that continues forever in one direction. A system can even leave only a line segment or a single point.
Real situations sometimes need more rules than a graph can show clearly. A school event plan could limit total spending, require a minimum number of tickets, and restrict the number of volunteers. The graph shows the allowed combinations, but the situation may add conditions that points must use whole numbers.
A graph can include a point representing two and a half volunteers, yet that value makes no sense in practice. Students should distinguish the mathematical region from the realistic answers within it.
Check labels on both axes, keep boundary styles consistent, and verify corners by substitution. A neat graph is helpful, but the original inequalities remain the final test for every answer.
Key Facts
- A linear inequality in two variables can be written as y < mx + b, y > mx + b, y <= mx + b, or y >= mx + b.
- Use a dashed boundary line for < or > because points on the line are not included.
- Use a solid boundary line for <= or >= because points on the line are included.
- The solution set of one linear inequality is a half-plane.
- The solution set of a system is the intersection of all the half-planes.
- A point (x, y) is a solution only if it makes every inequality in the system true.
Vocabulary
- Linear inequality
- A statement that compares two linear expressions using <, >, <=, or >=.
- Boundary line
- The line that separates the coordinate plane into regions for a linear inequality.
- Half-plane
- One side of a boundary line that contains the points satisfying or not satisfying an inequality.
- Feasible region
- The overlapping region that contains all points satisfying every inequality in a system.
- Test point
- A point substituted into an inequality to decide which side of the boundary line should be shaded.
Common Mistakes to Avoid
- Using a solid line for < or > is wrong because strict inequalities do not include points on the boundary line.
- Shading the wrong side of the boundary line is wrong because the solution must be the side where test points make the inequality true.
- Treating the union as the solution is wrong because a system uses the overlap where all inequalities are true at the same time.
- Forgetting to check boundary points is wrong because points on solid boundary lines may be solutions, while points on dashed boundary lines are not.
Practice Questions
- 1 Graph the system y >= 2x - 1 and y < -x + 5. Identify the feasible region and state whether the point (2, 3) is a solution.
- 2 For the system x + y <= 6, x >= 1, and y >= 2, find three ordered pairs with integer coordinates that are solutions.
- 3 A school club can spend at most 10 each and flyers cost $2 each, and the club wants at least 20 total items. Explain what the feasible region represents in this situation.