5.6 Graphing Systems of Linear Inequalities

Graphing Systems of Linear Inequalities

When you have more than one condition to satisfy at the same time, a single inequality isn't enough. A system of linear inequalities lets you graph multiple constraints on the same coordinate plane and find the region where all of them are true at once.

Solutions for Linear Inequality Systems

A system of linear inequalities is two or more linear inequalities that share the same variables. An ordered pair $(x, y)$ is a solution to the system only if it makes every inequality in the system true.

To check whether a point is a solution:

1. Substitute the $x$ and $y$ values into the first inequality. Simplify and check whether the statement is true.
2. Repeat for each remaining inequality in the system.
3. If the point satisfies all of them, it's a solution. If even one is false, it's not.

Example: Consider the system $y > 2x - 1$ and $y \leq x + 3$. Is $(2, 4)$ a solution?

• First inequality: $4 > 2(2) - 1 \rightarrow 4 > 3$ ✓
• Second inequality: $4 \leq 2 + 3 \rightarrow 4 \leq 5$ ✓

Both are true, so $(2, 4)$ is a solution to the system.

The solution set is the collection of all ordered pairs that satisfy every inequality in the system. On a graph, this shows up as a region, not just a single point.

Graphing of Inequality Systems

To graph a system, you graph each inequality on the same coordinate plane and then look for where the shaded regions overlap. Here's the process:

1. Rewrite each inequality in slope-intercept form ($y = mx + b$) if it isn't already.

2. Draw the boundary line using the slope $m$ and y-intercept $b$.

   • Use a dashed line for strict inequalities ($<$ or $>$), because points on the line are not included.
   • Use a solid line for inclusive inequalities ($\leq$ or $\geq$), because points on the line are included.

3. Shade the correct side of the line:

   • Shade above the line for $y > mx + b$ or $y \geq mx + b$.
   • Shade below the line for $y < mx + b$ or $y \leq mx + b$.
   • If you're unsure which side to shade, pick a test point (like $(0, 0)$ if it's not on the line), plug it in, and shade the side that makes the inequality true.

4. Repeat steps 1–3 for every inequality in the system.

5. Identify the overlapping region. The area where all shadings overlap is your solution set.

The solution region can be bounded or unbounded:

• A bounded region is fully enclosed by boundary lines, forming a shape like a triangle or polygon with a finite area.
• An unbounded region extends infinitely in at least one direction because the boundary lines don't close it off.

Visualizing Solutions on the Coordinate Plane

The coordinate plane makes it possible to see the solution set rather than just calculate it. Each boundary line represents one constraint, and the shading on each side shows which values satisfy that constraint.

Where the shaded regions intersect is the key part of the graph. Every point inside that overlapping region is a solution to the entire system. Points that fall in only one shaded region (but not the overlap) satisfy one inequality but not the other, so they're not solutions to the system.

When you're reading a graph, pay attention to whether boundary lines are dashed or solid. A point sitting right on a dashed boundary line is not part of the solution set, even if it's on the edge of the overlapping region.

Real-World Applications of Inequalities

Systems of linear inequalities show up whenever a problem involves multiple constraints at the same time. For example, suppose a bakery makes cakes and cookies. They have a limited number of oven hours and a limited budget for ingredients. Each product uses different amounts of each resource. The inequalities represent those limits, and the overlapping region on the graph shows every possible combination of cakes and cookies they could produce without exceeding either constraint.

To set up and solve a real-world problem:

1. Define your variables. Decide what $x$ and $y$ represent in the context of the problem (e.g., $x$ = number of cakes, $y$ = number of cookies).
2. Write an inequality for each constraint. Translate the limits (budget, time, materials, minimum requirements) into mathematical inequalities.
3. Graph the system on a coordinate plane to find the feasible region, which is the set of all combinations that satisfy every constraint.
4. Interpret the results. Use the feasible region to answer the question being asked, whether that's finding a specific combination or identifying the range of possibilities.

Note that many real-world problems also include the constraints $x \geq 0$ and $y \geq 0$, since quantities like hours worked or items produced can't be negative. These inequalities restrict the feasible region to the first quadrant of the coordinate plane.