Lesson Video: Two-Variable Linear Inequalities Mathematics • First Year of Secondary School

Video Transcript

In this video, we will learn how to graph two-variable linear inequalities. You should be familiar with the four different inequalities. We have greater than, greater than or equal to, less than, or less than or equal to.

Let’s start with a recap of how we graph a one-variable, or single-variable, inequality. We can take the example of 𝑦 is greater than two. If we demonstrated this on a number line, we would have a hollow circle above two and an arrow pointing to the right to indicate all the values which are greater than two. The fact that this circle on a number line is hollow or empty is really important. If it was colored in or filled in, then that would indicate the inequality greater than or equal to in this occasion. When the inequality is a strict inequality, like greater than or less than, then we would fill in the circle.

We have to consider this when we are graphing as well. When it comes to representing an inequality on a graph, we begin by temporarily replacing the inequality with an equals sign. So here we start by considering 𝑦 equals two. This can be shown on the graph, the point where 𝑦 is equal to two. But before we draw the line representing 𝑦 equals two, we need to consider the inequality. Because this inequality 𝑦 is greater than two is a strict inequality, then the line that we draw will be a dotted or dashed line.

And then, finally, we will need to represent the region where 𝑦 is greater than two. If we considered a point or coordinate below the line, let’s say this coordinate zero, one, at this point the 𝑦-value is one. But we know that one is less than two. So this wouldn’t be in the region where 𝑦 is greater than two. Taking another coordinate, this time the coordinate one, four which is above the line, the 𝑦-value at this point is four. Four is greater than two. And so this is part of the region where 𝑦 is greater than two, and so we can represent 𝑦 is greater than two by shading in the region that we wish. But a word of warning, sometimes in examinations, we’re asked to shade the region that we want and sometimes we’re asked to shade the region that we don’t want. So we must always be careful to read the question.

Graphing two-variable linear inequalities can be done by using exactly the same process. The dotted line represents the strict inequalities, either greater than or less than. But the complete full line represents the weak inequalities, greater than or equal to or less than or equal to. The difference is that instead of having just one variable of 𝑥 or 𝑦, we will have equations given in the different forms of the equation of a straight line, for example, 𝑎𝑥 plus 𝑏𝑦 equals 𝑐 or 𝑦 equals 𝑚𝑥 plus 𝑏. Because we are working with graphing equations such like this, then we should be confident with finding the equation of a straight line from its graph. We can now look at the first example. In this example, we will identify the correct notation for an inequality which has been represented on a graph.

Consider the inequalities shown in Figure 1 and Figure 2. Which of the following is correct? Option (A) Figure 1 shows 𝑦 is greater than 𝑥; Figure 2 shows 𝑦 is greater than or equal to 𝑥. Or option (B) Figure 1 shows 𝑦 is greater than or equal to 𝑥; Figure 2 shows 𝑦 is greater than 𝑥.

The first thing we need to do when we are identifying a graphed inequality is to work out the equation of the line, whether that line is dotted or a complete line. Both the lines given in Figure 1 and Figure 2 would represent the same equation. On both graphs, we can identify the coordinates one, one; two, two; three, three, and so on. Because every 𝑦-value is equal to the 𝑥-value, then the equation of either of these lines can be given as 𝑦 equals 𝑥. We notice that in both graphs, that is, in both figures, it’s the region above the line which has been shaded. The difference is that in Figure 1 we have a dotted line and in Figure 2 we have a complete line.

Dotted lines indicate a strict inequality, so they will either be greater than or less than, whereas complete lines represent weak inequalities. It will either be greater than or equal to or less than or equal to. So Figure 1 will have the graph of 𝑦 is greater than 𝑥 or 𝑦 is less than 𝑥. Figure 2 will represent 𝑦 is greater than or equal to 𝑥 or 𝑦 is less than or equal to 𝑥. Let’s take Figure 1 and pick a coordinate which is in the shaded region. So let’s pick the coordinate zero, four. The 𝑦-value of this is four, and the 𝑥-value is zero. The inequality that we would use between four and zero is that four is greater than zero. Because this coordinate is in the shaded region, then the region must be representative of 𝑦 is greater than 𝑥.

In the same way, we could pick the same coordinate in Figure 2, the coordinate zero, four. We know that four is greater than zero, but this time we know that it has to be a weak inequality: four is greater than or equal to zero. This graph represents the inequality 𝑦 is greater than or equal to 𝑥. We can therefore highlight that it’s the statement in option (A) which is correct. Figure 1 shows 𝑦 is greater than 𝑥; Figure 2 shows 𝑦 is greater than or equal to 𝑥.

Let’s look at another example.

Which inequality has been graphed in the given figure?

When we have an inequality represented on a graph, we will always have a line, whether that’s a complete line or a dotted line, along with a shaded region. The first thing we will need to do is identify the equation of the straight line. We can remember that the equation of a straight line can be given as 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 represents the slope, or gradient, and 𝑏 is the 𝑦-intercept. The 𝑦-intercept is usually very easily identified from the graph. It’s the point where the line crosses the 𝑦-axis. On this graph, this happens at the coordinates zero, negative three. And so the 𝑦-intercept is equal to negative three. The slope of a line can be found by the rise over the run. And we can work this out using two coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two by the slope is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one.

To make this process easier for ourselves, we should select coordinates which are easily identifiable. Usually, these will have integer values. We can see that the coordinates zero, negative three lie on the line but so do the coordinates one, one. It doesn’t matter which coordinates we designate with the 𝑥 one, 𝑦 one or 𝑥 two, 𝑦 two values. So let’s take 𝑥 one, 𝑦 one to be the coordinates zero, negative three. The slope will therefore be equal to one minus negative three over one minus zero. This simplifies to four over one, which of course is equal to four. Since we know that the slope 𝑚 is four and the 𝑦-intercept is negative three, we have the equation of the line as 𝑦 equals four 𝑥 minus three.

However, this isn’t the final answer since we were asked to write the inequality represented by the shaded region. Instead of the equals sign, we will need one of the inequalities, greater than, greater than or equal to, less than, or less than or equal to. So how do we know which one of these it will be? Well, we can notice that the line that we were given is a dotted line. This means it is a strict inequality. It will either be greater than or less than. The two other inequalities that involve equals to will not be a possibility.

We can now test which inequality it will be by selecting a coordinate in the shaded region, which does not lie on the line. Let’s pick the coordinate three, two. At these coordinates, the 𝑥-value is three and the 𝑦-value is two. We will therefore need to compare the value of 𝑦, which is two, with the value of four 𝑥 minus three, which is four times three minus three. When we simplify the right-hand side, we get nine. We, of course, know that two is less than nine. And therefore, the missing inequality must be less than. The answer is therefore that the inequality which has been graphed is 𝑦 is less than four 𝑥 minus three.

It’s worth pointing out that everything which is not in the shaded region and not on the line represents the inequality 𝑦 is greater than four 𝑥 minus three. If we checked this using a coordinate, for example, the coordinate zero, zero, plugging this into the inequality we would have 𝑦 is equal to zero and 𝑥 is equal to zero. On the left-hand side of the inequality, we would have zero. And on the right-hand side, we would have negative three. Since this region is 𝑦 is greater than four 𝑥 minus three, then the region which is shaded in is 𝑦 is less than four 𝑥 minus three.

We can now look at one final example.

Determine the inequality whose solution set is represented by the colored region.

The solution set of an inequality is the set of all points that satisfy that inequality. In this figure, this corresponds to all the points in the colored or shaded region. The first thing we will do is identify the equation of this straight line, which is the boundary line between the colored and the uncolored region. We sometimes use the equation of a straight line in the form 𝑦 equals 𝑚𝑥 plus 𝑏 to help us do this. However, sometimes there are graphs which are not that easy to read, and this is one of them. The 𝑦-intercept, the point where the line crosses the 𝑦-axis, is not particularly clear. It could be negative 1.5, but we can’t say for certain.

In cases like these, it can be useful to recall the point–slope form of the line given by 𝑦 minus 𝑦 one equals 𝑚 times 𝑥 minus 𝑥 one, where the coordinates 𝑥 one, 𝑦 one is a point on the line. The value of 𝑚 indicates the slope of the line. Let’s calculate the slope first. To calculate the slope between two coordinates 𝑥 sub one, 𝑦 sub one and 𝑥 sub two, 𝑦 sub two. We calculate 𝑦 sub two minus 𝑦 sub one over 𝑥 sub two minus 𝑥 sub one. When we are selecting two coordinates to use for this formula, it’s helpful to find coordinates with integer values. Do be careful with this graph because the major grid lines represent two on the 𝑥- and 𝑦-axis. But the minor grid lines, that’s the smaller grids, represents one-half.

One coordinate that we could select is the coordinate negative three, negative six. Another coordinate is that of one, zero. We can select the coordinates one, zero to have the 𝑥 sub one, 𝑦 sub one values and the coordinates negative three, negative six with the values of 𝑥 sub two and 𝑦 sub two. The slope of this line can therefore be calculated as negative six minus zero over negative three minus one. When we simplify this, we get negative six over negative four, which in turn can be simplified to three over two. The slope of the line 𝑚 is equal to three over two.

We can substitute this value into the point–slope form of the equation of a line, along with any coordinates on the line. We can use the same coordinates of one, zero in this equation. This will give us 𝑦 minus zero is equal to three over two times 𝑥 minus one. If we wanted to simplify this equation even further, we can multiply both sides of the equation by two. This gives us two 𝑦 equals three times 𝑥 minus one. Then distributing the parentheses on the right-hand side would give us two 𝑦 equals three 𝑥 minus three.

Any of these three equations would be valid equations of the straight line. However, we also need to identify the inequality represented by the shaded region. Let’s clear some space and work with this equation, two 𝑦 equals three 𝑥 minus three. Because we have an inequality, that means that instead of an equals sign, we’ll have one of the following: greater than, less than, greater than or equal to, or less than or equal to. We therefore recall that we need to check if this boundary line, the line of the equation, is dotted or a complete line. Since the line here is a dotted line, that means we’ll have a strict inequality. It will either be greater than or less than but none of the inequalities which involve the equal to portion.

In order to identify which one of these two inequalities it will be, we can check a coordinate which lies in the region which is represented. We could pick any coordinate in the region, but let’s pick a nice easy coordinate. The coordinate zero, zero lies in the shaded region. Remember that this means that the 𝑥-value is zero and the 𝑦-value is zero. So let’s substitute these values in. That means we’ll be comparing two times zero with three times zero minus three. When we simplify this, we’re comparing zero and negative three. Which inequality would we select here? Well, zero is greater than negative three. That means that the region must be two 𝑦 is greater than three 𝑥 minus three.

We can note that there are a number of different ways in which we can represent this inequality. For example, we could have subtracted three 𝑥 from both sides of this inequality. Or indeed, we could have collected all three terms on one side of the inequality. Any alternative forms are acceptable, provided we perform the correct steps when rearranging. So here we can give the answer that the inequality represented by the colored region is two 𝑦 minus three 𝑥 is greater than negative three.

We can now summarize the key points of this video. Firstly, we saw that the solutions to two-variable linear inequalities can be represented using graphs. Then, to determine the inequality that has been graphed, we first determine the equation of the boundary line using either the point–slope or slope–intercept forms of the equation of a straight line. The boundary lines of strict inequalities are broken or dashed lines. Weak inequalities are represented by solid lines. Finally, to determine the inequality sign, we consider the coordinates of a point in the shaded region. By substituting these coordinates into each side of the equation of the line, we can determine which side has the greater value and hence the inequality sign which we need.