 So let's solve the inequality 7x plus 3 greater than or equal to negative 9 AND 7x plus 3 less than or equal to negative 5. We'll use the test point method, and since this isn't AND inequality, a solution must satisfy both inequalities. So we'll solve the first inequality, we'll temporarily ignore the inequality and solve the equation 7x plus 3 equals negative 9, and so our solution has critical values x equal negative 12 7s. But, like a good math student or a good human being, we have to address the existence of the inequality. Now since the first inequality allows for equality, x equals negative 12 7 solves the first inequality, but we need to check to see if it solves the second inequality. So we'll check x equals negative 12 7s and see if it solves the second inequality. And since negative 9 is in fact less than or equal to negative 5, then x equals negative 12 7s also solves the inequality, and so this critical value is included. Now we have our second inequality, so again we'll solve it by ignoring the inequality, then addressing the existence of the inequality. Again, since our second inequality allows for equality, x equals negative 8 7 solves the second inequality, so it's a critical value, but we need to check to see if it solves the first inequality. So checking x equals negative 8 7s, we see that negative 5 is in fact greater than negative 9, this is true, and so our critical value at negative 8 7s is also included in our solution. Now let's graph the critical values, so we'll draw our number line, and our two critical values, negative 12 7s and negative 8 7s, are both included, so we'll graph them by using closed circles. And notice that the critical values separate the number line into three parts, so we need to test a point in each part. So on the left we'll test x equals minus 1 million, since this is an AND inequality, both inequalities must be satisfied in order for something to be part of the solution. So let's see if x equals minus 1 million satisfies that first inequality. We find, now notice that the left hand side is going to be a large negative number, which will not be greater than or equal to negative 9, so this is false, and so the left side will not be part of the solution. So we need to find something in the middle, and again, how you speak influences how you think. We want to find something between minus 12 7s and minus 8 7s, and so something between the two might be minus 10 7s. So we'll see if minus 10 7s satisfies both inequalities. So substituting that into the first inequality we find, we get a true statement, and if we substitute minus 10 7s into the second inequality we find, we also get a true statement. And since both inequalities are true, the middle interval is part of the solution. And on the right hand side we see that zero is included in the interval, so we'll test it out. We find substituting this into the first inequality, and this statement is true. But remember this is an AND inequality, so that means both inequalities must be true. So we do have to check that second inequality, but this is false, and since both inequalities need to be true, but the second is false, the right side will not be part of the solution. And so this allows us to write our answer in interval notation. We start at negative 12 7s included, and go up to negative 8 7s also included. And so we'll enter our answer. So remember the computer is stupid, and it only understands exactly what you tell it. So again, you can click on the preview to see what you've actually said, and triple check to make sure that what you've said is what you want to say, and only after you've made that confirmation click on submit.