4.7 Graphing Systems of Linear Inequalities

Graphing Systems of Linear Inequalities

A system of linear inequalities works like a system of linear equations, but instead of finding a single intersection point, you're finding an entire region of points that satisfy every inequality at once. This is how real-world constraint problems get modeled: budgets, production limits, and resource caps all translate into inequalities, and the overlapping region shows you what's actually possible.

Solutions of Linear Inequality Systems

A system of linear inequalities consists of two or more linear inequalities that share the same variables. Each inequality, when graphed, produces a boundary line and a shaded region on one side of that line. The solution set is the region where all the shaded areas overlap. Every point $(x, y)$ in that overlapping region satisfies every inequality in the system simultaneously.

The solution set takes one of two forms:

• Bounded (closed): The overlapping region forms a closed shape with finite area, like a triangle or quadrilateral. This happens when the inequalities "box in" the region from all sides.
• Unbounded (open): The overlapping region extends infinitely in at least one direction. For example, two inequalities might create a wedge that opens upward forever.

A point that sits on a boundary line is part of the solution only if that inequality uses $\leq$ or $\geq$. If the inequality is strict ($<$ or $>$), boundary points are not included.

Graphing a System of Inequalities

Here's the step-by-step process for graphing a system:

1. Rewrite each inequality in slope-intercept form ($y = mx + b$) if possible. This makes graphing easier because you can read the slope $m$ and y-intercept $b$ directly.

2. Draw the boundary line for each inequality.

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

3. Shade the correct side of each boundary line.

   • If the inequality is solved for $y$: shade above the line for $>$ or $\geq$, and below for $<$ or $\leq$.
   • If the inequality is solved for $x$: shade right of the line for $>$ or $\geq$, and left for $<$ or $\leq$.
   • Quick check: Pick a test point not on the line (the origin $(0, 0)$ is usually easiest). Plug it into the inequality. If it makes the inequality true, shade the side containing that point. If false, shade the opposite side.

4. Identify the overlapping region. The area where all shaded regions overlap is your solution set. Any point in that region satisfies every inequality in the system.

5. Express the solution set using set-builder notation when needed. For example:

$\{(x, y) \mid x \geq 0,\ y \geq 0,\ y \leq 2x + 1\}$

This reads: "all points $(x, y)$ such that $x$ is at least 0, $y$ is at least 0, and $y$ is at most $2x + 1$."

Components of Linear Inequalities

Each inequality in the system is built from a linear equation. The boundary line takes the familiar form $y = mx + b$, where:

• $m$ is the slope, describing how steeply the line rises or falls. A slope of $3$ means $y$ increases by 3 for every 1-unit increase in $x$.
• $b$ is the y-intercept, the point $(0, b)$ where the line crosses the y-axis.

You'll also use the x-intercept, the point $(x, 0)$ where the line crosses the x-axis. To find it, set $y = 0$ in the equation and solve for $x$. Plotting both intercepts gives you two points, which is all you need to draw the boundary line.

Real-World Applications

Many real-world problems involve constraints that translate directly into systems of inequalities. Here's how to set one up:

1. Define your variables. Decide what $x$ and $y$ represent. For example, $x$ might be the number of chairs produced and $y$ the number of tables.

2. Translate each constraint into an inequality. A budget cap of $500 with chairs costing $20 and tables costing $50 becomes $20x + 50y \leq 500$. A requirement to produce at least 5 chairs becomes $x \geq 5$.

3. Graph the system to find the feasible region, which is the set of all $(x, y)$ combinations that satisfy every constraint.

4. Interpret the results. Any point inside the feasible region is a valid solution. In optimization problems (like maximizing profit), the best solution typically occurs at a corner point of the feasible region. Check each corner to find which one gives the best outcome.