2.11.3 Sketching Solutions to Inequalities

Recognize the Relation Between the Solutions of an Inequality and its Graph

Now, we will look at how the solutions of an inequality relate to its graph. Let’s think about the number line shown previously again. The point $x=3$ separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See the figure below.

The solution of $x>3$ is the shaded part of the number line to the right of $x=3$

Similarly, the line $y=x+4$ separates the plane into two regions. On one side of the line are points with $y<x+4$ . On the other side of the line are the points with $y>x+4$. We call the line $y=x+4$ a boundary line.

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not $a$ is included in the solution:

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.

Now, let’s take a look at what we found in the example above. We’ll start by graphing the line $y=x+4$, and then we’ll plot the five points we tested, as shown in the graph. See the figure below.

In the previous example, we found that some of the points were solutions to the inequality $y>x+4$ and some were not.

Which of the points we plotted are solutions to the inequality $y>x+4$?

The points $(1,6)$ and $(-8,12)$ are solutions to the inequality $y>x+4$. Notice that they are both on the same side of the boundary line $y>x+4$.

The two points $(0,0)$ and $(-5,-15)$ are on the other side of the boundary line $y=x+4$, and they are not solutions to the inequality $y>x+4$. For those two points. $y<x+4$.

What about the point $(2,6)$? The point $(2,6)$ is a solution to the boundary line equation $y=x+4$, because $6=2+4$. However, $(2,6)$ is not a solution to the inequality, because the boundary line is not included in the solution to the inequality $y>x+4$.

Let's take another point above the boundary line and test whether or not it is a solution to the inequality $y>x+4$. The point $(0,10)$ clearly looks to above the boundary line, doesn't it? Is it a solution to the inequality?

$y>x+4$

$10>0+4$

$10>4$

So, $(0,10)$ is a solution to $y>x+4$.

Any point you choose above the boundary line is a solution to the inequality $y>x+4$. All points above the boundary line are solutions.

Similarly, all points below the boundary line, the side with $(0,0)$ and $(-5,-15)$, are not solutions to $y>x+4$, as shown in the figure below.

The graph of the inequality $y>x+4$ is shown in below.

The line $y=x+4$ divides the plane into two regions. The shaded side shows the solutions to the inequality $y>x+4$.

The points on the boundary line, those where $y=x+4$, are not solutions to the inequality $y>x+4$, so the line itself is not part of the solution. We show that by making the line dashed, not solid.

Try it

Recognize the Relation Between the Solutions of an Inequality and its Graph

The graph of $y=\frac{1}{2}x-3$ is shown here.

Use the graphing tool or technology outside the course. Graph the inequality that represents $y<\frac{1}{2}x-3.$ using the Desmos tool below.

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Compare your answer:

$y<\frac{1}{2}x-3$