Графіки лінійних нерівностей

Inequalities can have more than one variable. Here, we're going to take a closer look at inequalities that have two variables, $x$ and $y$.

When inequalities involve curves instead of lines, the process extends to graphing systems of nonlinear inequalities. We will learn about that in later lessons.

When you've got an inequality with more than one variable, the solutions you find will be ranges instead of single values. So, it's common to use graphs to show the range of solutions that solve the inequality.

Here is a graphic of a flowchart to help you better understand these steps:

Because of this, you will need to get your inequalities into slope intercept form (in other words, $y = mx+b$, but with an inequality sign instead of an equals sign).

Let's take a look at a few examples, starting with a pretty simple one:

$$y > x + 2$$

Let's find the slope and $y$-intercept.

The slope of the line is always $m$ in $y=mx + b$, and since there's no obvious answer for $m$ here we can conclude that it's $1$.

The $y$-intercept is always $b$ in $y=mx + b$. In this case, that's $2$!

Now we know that our slope is $1$ (coefficient of $x$) and our $y$-intercept is $2$.

Grab graph paper or a graphing calculator and try graphing this line on a coordinate plane!

There are two things we need to do now:

1. If the inequality symbol is $>$ or $<$, the line needs to be dotted. If the sign is $≤$ or $≥$, the line remains a solid line.
2. Shade the area of the graph that shows the range of solutions to the inequality.

Another important note: Even with linear inequalities, it is possible for a system to have no solution when shaded regions do not overlap.

Notice how we have a dotted line here because our inequality sign was $>$. The site of the boundary line is basically what we graphed earlier.

Note: if we had a greater than or equal to sign, $≥$, then the line would've been solid.

We also shaded the graph to show the range of solutions that make our inequality true. Recall that our inequality tells us that $y>x+2$.

This means that y is greater than the corresponding $x$ value plus $2$. For instance, let's say that $x=4$. This means that $y$ is greater than $6$ when $x=4$. Check the graph - the solution to the inequality, i.e., the values that satisfy the inequality, is the shaded area.

Let's look at another example:

$$2y + 4 < 6x$$

First, we need to get this inequality into slope intercept form.

$$2y < 6x - 4\\ \frac{2y}{2} < \frac{6x-4}{2}\\ y<3x-2$$

Now we need to graph this line.

Because we have $<$ in our inequality (instead of $≤$ or $≥$) our line will be dotted.

A Brief Introduction

How do you feel about graphing linear equations such as $y=3x-5$ and $6x-11y=21$? You'll need all the same skills for graphing linear inequalities, which can end up looking like this:

Part I - The Boundary Line

The first part of the basics is graphing the boundary line, which is the edge of the solution set. Depending on what kind of inequality you have, numbers on the boundary line will be in the solution or not.

For $>$ and $<$, the boundary line is not included in the solution set. You show this by making the boundary line dashed.

For $\le$ and $\ge$, the boundary line is included in the solution set. You show this by making the boundary line solid.

Remember that when graphing an equation in slope-intercept form ($y=mx+b$), the value $b$ is on the $y$-axis, and when using standard form, it's better to find the intercepts rather than the slope.

Part II - Shading the Solution Set

The easiest method to figure out which side of the graph to shade (showing where the solutions are) is to pick a point not on the line. Substitute its coordinates into the inequality. If the coordinates make the inequality true, shade the side of the graph the point is in; otherwise, shade the opposite side.

If you plug the coordinates into the equation you get:

$$\begin{align} y &\ge 2x + 3 \\ (2) &\ge 2(4) + 3\\ 2 &\ge 8 + 3\\ 2 &\ge 11 \end{align}$$

$2$ isn't greater than $11$, so $(4,2)$ must not be part of the solution set. That means that the opposite side of the line should be shaded!

(Video) Graphing Linear Inequalities

by Friendly Math 101

Here's a video by Friendly Math $101$ showing you how to graph linear inequalities. These are similar to graphing equations except there can be solid or dashed lines and there will also be a shaded region.

$\large{\text{Summary}}$

The first example she looks at is $y<\frac13x-2$. From first glance, we can see our slope is $m=\frac13$ and our $y$-intercept is $b=-2$. Next, the inequality sign, $<$, tells us that we will shade below the line. Also, the line will be dashed instead of solid because there's no equals to symbol underneath.

Once you have all the information you can begin by graphing the line, making sure that its dashed.

Finally, we shade in the region based on the inequality. If you are unsure you can always plug in a point not on the line into the inequality. If that side of the line is shaded, the point will give you a true statement, like $3=3$. If it's not shaded then it will give you a false statement, like $3=0$.

Try graphing this inequality on your own and then watch the video to see the answer! $$y\geq-\frac12x+2$$