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At a Glance - Inequalities

An inequality is just a type of relation, which means we can graph it like we would graph any relation. The trick with inequalities is that, instead of drawing lines to connect the dots, we have to shade in big areas of the graph. You could use some shade though. You're starting to burn.

Sample Problem

Graph the inequality yx.

Let's find some points in the relation. This relation will contain every ordered pair of real numbers in which y is at least as big as x.

We can plot the points we have so far:

We can see that every point along the line y = x will be included in the relation, since for any point on that line y is at least as big as x. Good news for all you line-lovers out there. Let's include this line in our picture:

We have some points already, which are all above the line. If we were to find more points between the ones we already have, we would find that they're all in the relation. The relation contains every point in or above the line y = x, which we show by shading in that half of the graph. You might need to take an extra trip to Staples. You're really going to start burning through your pencil supply.

A linear inequality is an inequality that can be written with y on one side and a linear polynomial in x on the other. Let's look at these for a bit, since they're easier to think about that other inequalities. We promise we'll try to make your brain hurt more later. To graph a linear inequality, we

  • pretend we have an equation and graph the line, and then
  • figure out which side of the line we want to shade.

We don't generally advocate taking sides, but in this instance it's perfectly acceptable.

Sample Problem

Graph the inequality y + 3 ≤ 2x.

The first thing we do is graph the line

y + 3 = 2x

which is also known as the line

y = 2x – 3.

Then we need to figure out which side of the line we want to be on. We can do that by picking a point on one side of the line, and seeing whether or not that point satisfies the inequality, i.e. whether that point is in the relation. Let's pick an easy point, like (1, 4). Sorry to refer to him as easy, but it's true. He's an undeniably gullible pushover.

When x = 1 and y = 4, our original inequality becomes:

4 + 3 ≤ 2(1)
7 ≤ 2

That's definitely not true—7 sure isn't smaller than 2—so this point does not satisfy the inequality. That means we shade in the other side of the graph:

When an inequality is strict (like, with a "<" or ">" symbol instead of "≤" or "≥"), we do the same thing as above except that now we don't want to include the points on the line. In fact, we don't want to do anything that might upset it, considering how strict it is. We are so over being grounded.

To show that we aren't including the points on the line, we draw a dashed line instead of a solid line. For example, if we wanted to graph y + 3 < 2x instead of y + 3 ≤ 2x, we'd use the exact same graph as above, except we'd draw a dashed line and shade everything underneath it.

Example 1

Graph the inequality 6x + 4y ≤ 1.

Example 2

Graph the inequality x + 1 > y.

Example 3

Find the inequality graphed below.

Exercise 1

Graph the following inequality: y ≥ 2x + 7.

Exercise 2

Graph the following inequality: y > -4x + 3.

Exercise 3

Graph the following inequality: x ≤ 4.

Exercise 4

Determine the inequality graphed in the picture below.

Exercise 5

Determine the inequality graphed in the picture below.

Exercise 6

Determine the inequality graphed in the picture below.

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