Graphing Linear Inequalities

1. Two-Variable Inequalities

Previously we learned to solve inequalities with only one variable:

EXAMPLE

Solve the inequality , graph, and write in interval notation.

	Divide both sides by -2
	Divide by a negative - flip symbol!
	Graph, starting at -3, going down with a closed circle for less than or equal to
	Interval Notation

But what about inequalities that contain two variables, such as ? In particular, we will look at linear inequalities in two variables which are very similar to the linear equations we have seen in this unit.

hint

Two-variable linear inequalities have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business made a profit.

2. Graphing a Two-Variable Inequality on the Coordinate Plane

When graphing a linear inequality, we generally follow these steps:

step by step

1. Analyze the inequality symbol to determine a solid or dashed line
2. Graph the equation of the line as using either a solid or dashed line
3. Determine a solution region to shade either above or below the line

Let's break down each step further:

1. Analyze the inequality symbol to determine a solid or dashed line: First, we need to decide if the graph of the linear inequality will have a solid or a dashed line. Solid lines indicate that the exact value of y is included in the solution to the inequality. We use solid lines when the inequality symbol is non-strict, or includes "or equal to." These would be our inequality symbols ≤ and ≥.

On the other hand, we use dashed lines when the exact value of y is not allowed in our solution to the inequality. Dashed lines represent strict inequalities, which do not include "or equal to." These are the symbols < and >.

Endpoint IS included in interval	Endpoint IS NOT included in interval
or	< or >

big idea

Use solid lines when graphing inequalities with the non-strict symbols ≤ and ≥. Use dashed lines when graphing inequalities with strict symbols < and >.

2. Graph the equation of the line as y = using either a solid or dashed line: Once we have determined what kind of line to use when graphing the inequality, we need to actually graph it. To do so, we somewhat ignore the fact that we are dealing with an inequality, and simply graph the line as if it were rather than with an inequality symbol. Just remember to use the appropriate line.

3. Determine a solution region to shade either above or below the line: We're not finished with the graph of our inequality yet. Next, we need to shade in half of the coordinate plane to represent the solution region. The solution region highlights all possible x- and y-coordinates that satisfy the inequality. When boundary lines are in the form (with the variable y isolated on one side), it is simple enough to once again examine the inequality symbol to see which portion of the coordinate plane to shade.

• If the inequality symbol is "greater than y" or "greater than or equal to y", we shade everything above the line.
• If the inequality symbol is "less than y" or "less than or equal to y", we shade everything below the line.

EXAMPLE

Graph the linear inequality .

Step	Explanation	Graph
1. Use solid or dashed line.	To start, we note that the inequality symbol is strict, thus we will use a dashed line.
2. Graph the equation of the line as .	Next, we note the y-intercept of -5 and the positive slope of 2 to create the start of our graph.
3. Shade above or below the line.	Since our inequality is , a less-than inequality, we know to shade everything below our dashed line.

This is the graph of our inequality. The dashed line tells us that the exact values of x and y that fall on the line are not included in our solution. The line is considered the boundary line, where coordinates on one side of the line are solutions, and coordinates on the other side of the line are non-solutions. The highlighted region is known as the solution region, as it shows all possible x- and y-values that satisfy the inequality.

3. Using a Test Point to Shade the Solution Region

While using the above method to determine which half of the coordinate plan to shade, there is a more mathematically sound method. This method will be particularly useful when graphing non-linear inequalities, where "above the line" and "below the line" aren't as clear cut.

A test point is any coordinate (x, y) that does not lie on the boundary line. We can think of it as being a representative for all other coordinates on that side of the boundary line. If our test point satisfies the inequality, it represents a solution, as do all the other points on that side. If our test point does not satisfy the inequality, then it represents a non-solution, and the other side of the boundary line should be shaded.

EXAMPLE

Consider the example from above. Suppose we have completed Step 2 and have drawn our boundary line. Now we need to determine which side to shade.



Let's use the test point (0, 0). This is a good strategy, so long as the origin is not on the line itself.

	Plug in the point (0, 0) into the inequality
	Multiply 2 and 0
	Subtract 5 from 0
	is a false statement, meaning (0, 0) is not in the solution region

We determined that the origin is not part of the solution region, because the point (0, 0) does not satisfy our inequality. This means that all points on the other side of the boundary line are solutions, and we should shade in that half-plane on the graph, as shown below.

hint

Choosing (0, 0) whenever possible is wise, because it is easy to multiply by 0 and add 0, making our calculations easier.

big idea

A test point is a coordinate point (x, y) that is a representative for all points on one side of the boundary line. Use the coordinates to plug into x and y in the inequality statement. If the statement is true, it represents a solution, as does that entire half-plane. If the statement is false, it represents a non-solution, and the other half-plane is the solution region.