 So let's try to solve the compound inequality 8y plus 16 less than 1 or 8y plus 16 greater than or equal to 17. We'll use the test point method. So we'll take our first inequality and ignore the inequality and pretend that it's equality and solve the equation 8y plus 16 equals 1. But like a good math student or a good human being, we have to acknowledge the existence of the inequality. And since the corresponding inequality is strict, the solution does not include this critical value. And so our inequality has critical value y equals negative 15 eighths, which is not included. The other inequality, 8y plus 16 greater than or equal to 17, so again we'll ignore the inequality and solve the corresponding equation 8y plus 16 equals 17, which gives a solution y equals 1 eighth. And again, like a good math student or a good human being, we acknowledge the existence of the inequality. Now, since the corresponding inequality allows for equality, this endpoint solves the second inequality. But remember this is a compound inequality. Fortunately, since the compound inequality is an or, only one inequality has to be true. And since we've solved the second inequality with y equals 1 eighth, this critical value is included in the solution. And so our critical value y equals 1 eighth is included. Let's graph those critical values on the number line. So we have our number line. y equals negative 15 eighths is here someplace on the negative side. y equals 1 eighth is over here someplace on the positive side. And since negative 15 eighths is not included, we'll put an open circle there. On the other hand, 1 eighth is included, so we should put a closed circle there. And the critical values will separate the number line into three parts. We'll test a point in each part. Since this is an or inequality, only one of the inequalities must be true. So on the left, we'll test y equals, well, let's go big, minus 1,000. So the question you've got to ask is, is the inequality 8y plus 16 less than 1 or 8y plus 16 greater than or equal to 17 is true for y equals minus 1,000. So we'll replace y with minus 1,000. And the left hand side is going to be a pretty big negative number, which will definitely be less than 1, and so the left interval is included. In the center, we'll test y equals 0. So we want to know whether our inequality 8y plus 16 less than 1 or 8y plus 16 greater than or equal to 17 is true for y equals 0. So we'll check the first inequality if y equals 0 we get, which isn't true. But since this is an or inequality, only one of the inequalities must be true, so maybe we'll get lucky, and the other inequality will be true. So we'll let y equals 0. And again, this is not true. And since neither of these are true, the center is not included. On the right side, we'll test y equals 1 million, and so we'll let y equals 1 million in our first inequality, and we find over on the left hand side, we have a very large number, which will not be less than 1. But again, since this is an or inequality, only one of the inequalities must be true. So let's see if the other one is true. So we'll let y equals 1 million. And now on the left hand side, we have a very large positive number, which will definitely be greater than or equal to 17, and so the right interval is included. And so our solution will consist of two intervals. On the left, everything from minus infinity up to minus 15 eighths, not the center, but on the right, everything starting at 1 eighth and going to infinity. And we write this in interval notation as... Now remember, computers are stupid, and they'll believe exactly what you tell them, and they'll do exactly what you ask them to do. And what that means is that even though we have the correct answer here, we've got to write it in a way that the computer understands. So remember, we'll use the OO to indicate infinity, and maybe we'll type in this answer. Now we do need to indicate the union of two intervals, and so we'll use the U to indicate that. And type in the second interval. The computer doesn't say a syntax error, so it understands what we've written. So let's hit submit. And we got little x that says this answer is wrong. So what happened? Well, if we click on the preview button, we'll see what we actually entered. And we see that what we actually told the computer is different from what we meant to tell the computer. And remember, no computer ever lost its job because of a typo. So let's try and fix our answer. Remember, plus or minus infinity is never included in an interval, so we should always have parentheses wherever we have those infinities. So we'll change those. So remember, my open math preview is what we've actually entered. So if I hit submit, it's going to grade this answer whether or not that's what I wanted to enter. So let's hit submit. And again, it marked the answer wrong. Again, to see why this happened, check the preview. And if we look carefully, we see that we used a square bracket instead of a parentheses. Now, it may seem that checking all this syntax is a tedious, pointless task, but actually this is a life skill. A misplaced comma or parentheses can cost a company millions of dollars and no computer ever got fired for a typo. So let's fix this. Again, my open math previews what we've actually entered, and so we should triple check to make sure that what we've written is what we wanted to write. And only if it is should we hit submit.