Video Transcript
Find the system of inequalities that forms the triangle shown in the graph.
We can recall that a system of inequalities is a list of different inequalities which form a particular region. When we’re given a region, like we have been in this question, the first thing to do is to find the equation of each line which forms a boundary of the region. In this question, we’ll need to find three different inequalities as we have three lines shown on the graph. The first line, which is not a dotted line, slopes downwards from left to right, and it passes through the origin. The second line slopes upwards from left to right. It’s a dotted line. And it goes through six on the 𝑦-axis. Finally, we have this third line, which is also not a dotted line. And it slopes downwards from left to right.
What we’ll do first is to find the equation of each of these three lines, and then we’ll think about the inequality that’s represented by the shaded part. Before we begin with line one, we’ll need to remember the general form of the equation of a straight line: 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope or gradient of the line and 𝑏 is the 𝑦-intercept. If we have a look at line one, let’s find the slope of this line. The slope of a line can be found by the rise over the run or the change in 𝑦 over the change in 𝑥. So looking at this triangle that we’ve drawn underneath line one, the change in 𝑦 will be negative one and the change in 𝑥 would be one. Therefore, the slope is negative one over one. And that’s equal to negative one.
Next, to find the 𝑦-intercept, we look for the place where it crosses the 𝑦-axis. And this will be at zero, zero. So the 𝑦-value will be zero. Now that we have the slope and the 𝑦-intercept, we can put this into the equation of the line. 𝑦 is equal to negative one 𝑥 plus zero. And of course, we can simplify this to give us 𝑦 is equal to negative 𝑥.
Now let’s have a look at finding the slope and 𝑦-intercept of line two. We can draw a triangle anywhere on this line to help us find the slope, but it’s often useful to find integer values for the 𝑥- and 𝑦-coordinates. The line passes through the coordinate negative two, two and negative one, four. This time, the change in 𝑦 or the rise will be two and the change in 𝑥 or the run will be one. So the slope is equal to two over one, which is two. It’s always a good idea to check, as this line slopes upwards from left to right, then we should have a positive slope, which we have indeed found.
Then we need to find the 𝑦-intercept of this line, so we look for the point where the line crosses the 𝑦-axis. As this happens when the 𝑦-value is six, then the 𝑦-intercept is six. Putting both of these pieces of information into the general equation of a straight line, we have that 𝑦 is equal to two 𝑥 plus six. We can now clear some space so that we can find the equation of line three. Let’s start once again by finding the slope of line three. This line goes through the point one, eight and also through two, negative two. The rise of this line will, therefore, be negative 10. And we can see that it’s eight units above the 𝑥-axis and two units below. The run will be one. So therefore, the slope, which is the rise over the run, can be written as negative 10 over one, which simplifies to just negative 10.
As a quick check of this value, this line is steeper than the other two lines, so we can expect the absolute value of the slope to be larger. And next, it’s sloping downwards from left to right, which would indicate that the slope will be a negative value. As we then go to find the 𝑦-intercept of this line, you might notice that we’ll have a problem. We can’t actually see where the line crosses the 𝑦-axis. So therefore, we’ll need to have another way to find the equation of this line.
This form of a line, 𝑦 minus 𝑦 sub one equals 𝑚 times 𝑥 minus 𝑥 sub one, is often referred to as the point slope form of a line. When we have the coordinate or ordered pair 𝑥 sub one, 𝑦 sub one and we know the slope 𝑚, then we can fill these values into this point–slope form to find the equation of the line. We’ve already established that the point one, eight lies on the line. So one and eight can be our 𝑥 sub one and 𝑦 sub one values. And we know that 𝑚, the slope, is equal to negative 10. So we can write out the point–slope form and fill in the values. 𝑦 minus eight is equal to negative 10 multiplied by 𝑥 minus one.
When we distribute the negative 10 across the parentheses, we’ll have negative 10𝑥. And then we’ll have negative 10 multiplied by negative one, which gives us a positive value of 10. We can isolate the 𝑦-variable on the left-hand side by adding eight to both sides of this equation. So we’ll have 𝑦 is equal to negative 10𝑥 plus 18. We have now found the three equations of these three lines. We can now take each line in turn in order to identify the region and how it compares to the equation of that line.
When we have a line that is not dotted, that will represent that the values can also lie on this line. In other words, we’re looking at the inequalities greater than or equal to or less than or equal to. The triangle or shaded region is above line one. So that represents all the values where 𝑦 is greater than negative 𝑥. And remember, we also need to include those values where 𝑦 is equal to negative 𝑥. And so that’s the first inequality. If the region had been shaded below this line, then the inequality would have been 𝑦 is less than or equal to negative 𝑥.
For the second line, we know that the equation is 𝑦 equals two 𝑥 plus six. But notice that this line is dotted, so our inequality will not include the values where 𝑦 is equal to two 𝑥 plus six. The shaded region is below this line, so it represents all the values where 𝑦 is less than two 𝑥 plus six. And that’s our second inequality. Finally, we have the third line with the equation 𝑦 is equal to negative 10𝑥 plus 18. The triangle and shaded region is below this line, but is it going to be a less than or a less than or equal to sign?
Well, as we have a complete line and not a dotted line, that means we also need to include those values where 𝑦 is equal to negative 10𝑥 plus 18. So that’s our third inequality. We can, therefore, give the answer that this system of inequalities are the inequalities 𝑦 is less than two 𝑥 plus six, 𝑦 is greater than or equal to negative 𝑥, and 𝑦 is less than or equal to negative 10𝑥 plus 18. And it doesn’t matter which order we write those inequalities.
