What is the inequality graphed in the given figure?
Before we try and work out what this inequality is, we’re gonna quickly recap the inequality signs. First of all, we’ll start with 𝑎 is greater than 𝑏. Next, we have 𝑎 is less than 𝑏. Our third inequality sign has a slight subtle difference. It actually means 𝑎 is greater than or equal to 𝑏. And you can notice this by the line, which is below the inequality sign.
But what’s the difference of this when we’re looking at it practically? Well, here’s a quick example which identifies what the difference is. We take a look at this inequality. We have 𝑥 is greater than four. So what this means is any value that is higher or more than four would be included in this inequality. But the key is it would not include four itself.
However, the second inequality 𝑥 is greater than or equal to four would mean that any value greater than four, but also including four would be included in the inequality. So that’s the difference: the bottom inequality would include four; the top inequality would not. And finally, our inequality is 𝑎 is less than or equal to 𝑏.
Now, we’ve recapped our inequality signs, we can actually get on with trying to solve what inequality has been shown on the graph. If we look at the graph, we can see that the shaded area represents everything that is above this line. And I’m showing that with arrows. However, what is the line that it’s above? Well, this horizontal line is actually 𝑦 is equal to two cause we can see that all the 𝑦-coordinates along there would have a value of two.
So great! We can now say that the shaded area that we’re looking for is 𝑦 is greater than two. So is that it? Have we finished? Well, no because if we look a bit closer, we can say okay 𝑦 is greater than two. But it could also be 𝑦 is greater than or equal to two. Which one of these is it? Well, this is where we have to introduce ourselves to a bit of notation that’s used when we’re graphing inequalities.
Well, first of all, if we look, a solid line would indicate that a value is included, whereas a dashed or dotted line indicates that the value is not included. Okay, so this extra bit of information can help us decide which inequality it is.
Well, looking back at our graph, we can see yes, it is a solid line. So therefore, two is included in our inequality. And this can help us eliminate one of our inequalities. We can eliminate our top inequality that says that 𝑦 is greater than two because this would not include two. But as it’s a solid line, we can see that we must include two. So therefore, our bottom inequality 𝑦 is greater than or equal to two is the inequality graphed in the given figure.
And now, we’ve solved the problem. It’s just key to remember be careful of that solid or dashed line because often that is the key and most common mistake that could be made in a problem like this.