Solve the absolute value of 𝑥 minus six is less than or equal to five.
So the first thing you notice in this question is these vertical lines. And these vertical lines mean the absolute value or modulus of 𝑥 minus six. So if we think about this as the absolute value of 𝑎 or the modulus of 𝑎. What does this actually mean? But what it means is the distance between 𝑎 and zero. So therefore, it doesn’t matter if it’s positive or negative because it will still be the same distance away from zero.
So therefore, if we actually have something like the modulus of 𝑥 or the absolute value of 𝑥 is equal to 𝑎 as an equation, then this would in fact have two solutions: 𝑥 is equal to 𝑎 or 𝑥 is equal to negative 𝑎 because they’re both the distance 𝑎 from zero. Okay, so now we have an understanding of what the absolute value or modulus is. How can we use it to solve our inequality? Well before we can solve a question involves an inequality, we’ve got to look at how an inequality would work with a modulus sign.
So let’s consider the modulus or the absolute value of 𝑥 is less than three. But what this means is all the points that are less than three units away from zero. So let’s look how this would look on our number line. Well first of all, you can see that we’ve got open circles. And this is because it’s less than. It’s not less than or equal to. If it’s less than or equal to or greater than or equal to, they’d be colored in. But what this shows is all the values that are less than three units away from 𝑥 go from negative three up to three cause any value within this is less than three units away from zero.
So this can be written as 𝑥 is greater than negative three or less than three. So this is us saying that the modulus or the absolute value of 𝑥 is less than three can be written in this way as a compound inequality. Okay, great! So this gives us something to work with that we’re gonna try and solve. The modulus or absolute value of 𝑥 minus six is less than or equal to five because we can rewrite our modulus or absolute value in inequality as 𝑥 minus six is greater than or equal to negative five, but less than or equal to five. And we can do that because this is the less than inequality as per the one in our example.
It does work slightly differently if it’s a greater than inequality. So now in order to solve this inequality, we’re gonna add six to each section. So we’re gonna add six to negative five, six to 𝑥 minus six, and six to five. So when we do that, our first value is going to be one because negative five add six is one, then our next values going to be 𝑥 because 𝑥 minus six add six is just 𝑥, then finally we’re gonna to have 11 because five add six is eleven. Therefore, we can say that the solution to the modulus of 𝑥 minus six is less than or equal to five is 𝑥 is greater than or equal to one but less than or equal to 11.