System Of Inequalities

What Is a System of Linear Inequalities?

A system of linear inequalities is a collection of two or more linear inequalities that share the same variables. While a single linear inequality like $y>2x+1$ has an infinite number of solutions represented by a shaded region on a graph, a system of inequalities asks us to find all the coordinate pairs $(x,y)$ that make all of the inequalities in the system true at the same time.

Think of it like planning a party. You have two rules (inequalities):

• You can spend no more than $50 on snacks.
• You must invite at least 10 people.

A successful party plan (a solution) must satisfy both conditions. In algebra, our conditions are inequalities, and our solutions are points on a coordinate plane. The solution to the system is the set of all points that lie in the overlapping shaded regions of each individual inequality. This overlapping area is often called the feasible region or the solution set.

How Do You Graph a Single Linear Inequality?

Before we can tackle a whole system, we need to be experts at graphing just one linear inequality. It's a straightforward process if you follow these steps. Let's use the inequality $y<3x-1$ as our guide.

1. Graph the Boundary Line: First, pretend the inequality symbol is an equals sign and graph the corresponding linear equation. For $y<3x-1$, we graph the line $y=3x-1$. You can do this by finding the y-intercept (at $y=-1$) and using the slope (rise 3, run 1) to find more points.
2. Choose a Solid or Dashed Line: This is a crucial step. The type of line you draw depends on the inequality symbol. A solid line means the points on the line are included in the solution, while a dashed line means they are not.

   Symbol	Meaning	Line Type
   $<$ or $>$	Less than / Greater than	Dashed
   $\le$ or $\ge$	Less than or equal to / Greater than or equal to	Solid

   Since our example is $y<3x-1$, we use a dashed line.

3. Pick a Test Point: Choose any point on the coordinate plane that is not on the boundary line. The easiest point to use is almost always the origin, $(0,0)$, unless the line passes directly through it.
4. Test and Shade: Substitute the coordinates of your test point into the original inequality.
   • If the test point makes the inequality true, you shade the entire region on the side of the line that contains the test point.
   • If the test point makes the inequality false, you shade the other side of the line.

   Let's test $(0,0)$ in $y<3x-1$:
   $0<3(0)-1$
   $0<0-1$
   $0<-1$
   This statement is false. Therefore, we shade the side of the dashed line that does not contain the origin $(0,0)$.

How Do You Solve a System of Inequalities by Graphing?

Solving a system is just an extension of graphing a single inequality. You perform the same steps, but you do it for each inequality on the same coordinate plane. The solution to the system is the area where the shaded regions from all the inequalities overlap.

Here is the step-by-step process:

1. Graph the first inequality, including its boundary line (solid or dashed) and shading. It can be helpful to use a light color or diagonal lines in one direction.
2. On the same set of axes, graph the second inequality, including its boundary line and shading. Use a different color or draw your shading lines in a different direction (e.g., perpendicular to the first).
3. Identify the region where the shadings overlap. This region of intersection is the solution set to the system. Every single point in this overlapping area satisfies all inequalities in the system.
4. If there is no region where the shadings overlap, the system has no solution.

What Does the Solution to a System of Inequalities Look Like?

Unlike a system of linear equations which often has a single point of intersection as its solution, the solution to a system of inequalities is a region. This region contains an infinite number of points, and every single one of them is a valid solution.

To verify your solution, you can pick any point from the overlapping shaded region and plug its coordinates back into all of the original inequalities. The coordinates must make every inequality a true statement.

What About Special Cases Like 'No Solution'?

Most systems you encounter will have a clear, overlapping solution region. However, there are a couple of special cases you should be aware of.

No Solution

Sometimes, the shaded regions of the inequalities will never overlap. When this happens, there is no point $(x,y)$ that can satisfy both inequalities at the same time. This system is said to have no solution.

This often occurs with parallel boundary lines. For example, consider the system:

Both lines have a slope of $1$, making them parallel. The first inequality requires shading above the line $y=x+4$, while the second requires shading below the line $y=x-1$. Because the shaded regions are on opposite sides of these parallel lines, they will never intersect. There is no overlap, and thus no solution.

Unbounded Regions

An unbounded region is a solution set that extends infinitely in at least one direction. All of the examples we've looked at so far have unbounded regions. This is not a 'special' case in the sense of being rare, but it's important to recognize that the solution isn't always a neat, enclosed polygon. The solution can be an entire half-plane, a quadrant, or a wedge that goes on forever.

How Are Systems of Inequalities Used in the Real World?

Systems of inequalities are extremely useful in a field called linear programming, which is used to find the best possible outcome in a given situation. This is often used in business to maximize profit or minimize cost based on a set of constraints.

What Are Common Mistakes to Avoid?

• Mixing up Solid and Dashed Lines: This is the most frequent error. Remember the rule: if the symbol includes 'or equal to' ($\le ,\ge$), the line is solid. If it's strictly less/greater than ($<,>$), the line is dashed.
• Shading the Wrong Side: Always, always use a test point! Don't just guess which side to shade based on whether the symbol is 'greater than' or 'less than'. This can be misleading, especially if the inequality isn't in slope-intercept form. The test point method is foolproof.
• Graphing the Boundary Line Incorrectly: Double-check your slope and y-intercept before drawing your line. A small error in the line will cause your entire shaded region to be incorrect.
• Forgetting to Flip the Symbol: When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. Forgetting this will lead you to shade the wrong region. For example, $-2y<4x+6$ becomes $y>-2x-3$.
• Stopping at the Intersection Point: For a system of equations, the solution is the point of intersection. For inequalities, the intersection point of the boundary lines is just one part of the boundary; the solution is the entire overlapping region.

Quick Summary: The Step-by-Step Process

Feeling overwhelmed? Just follow these steps every time you need to solve a system of linear inequalities.

1. Isolate y: For each inequality, rewrite it in slope-intercept form ($y=mx+b$). Be careful to flip the inequality symbol if you multiply or divide by a negative number.
2. Graph the First Boundary Line: Draw the line on the coordinate plane. Make it solid for $\le$ or $\ge$ and dashed for $<$ or $>$.
3. Shade for the First Inequality: Pick a test point (like $(0,0)$) and substitute it into the inequality. If it's true, shade the side with the point. If false, shade the other side.
4. Graph the Second Boundary Line: On the same plane, draw the second line, again paying attention to solid vs. dashed.
5. Shade for the Second Inequality: Use a test point for the second inequality and shade its solution region, preferably with a different color or pattern.
6. Find the Solution: The solution to the system is the region where your two shaded areas overlap. Clearly mark this final solution region.