Find the solution set of the inequality negative four is less than or equal to negative eight minus 𝑥, which is less than two squared in all real numbers. Give your answer in interval notation.
Starting with this inequality, we need to isolate 𝑥 in the middle. We want this middle section to be 𝑥 by itself, and we want that 𝑥 value to be positive. However, before we even start that, let’s go ahead and square two. Two squared equals four. The first thing we do to get 𝑥 by itself is to add eight. We have a negative eight plus eight will equal zero. And to keep this inequality equivalent, we need to add eight on either side of the inequality. Negative four plus eight equals four. Less than or equal to doesn’t change. Inside the inequality, negative 𝑥 is left and then we have four plus eight, which is 12. We want to turn this negative 𝑥 into positive 𝑥. To do that, we either multiply or divide by negative one. Negative 𝑥 divided by negative one equals 𝑥, but if we divide by negative one in one part of the inequality, we do it in all parts of the inequality. Four divided by negative one is negative four; 12 divided by negative one equals negative 12.
Now this is an important step. When we are multiplying or dividing by negative and we have inequalities, our less than or equal to becomes greater than or equal to and our less then becomes greater than. This says that negative four is greater than or equal to 𝑥, and 𝑥 is greater than negative 12. We can imagine this on a number line. If we have negative 12 and negative four on the number line, we know that negative four is greater than or equal to 𝑥. That’s the same thing as saying 𝑥 is less than or equal to negative four. 𝑥 can be equal to negative four, and it’s less than negative four. However, the second half of the inequality says that 𝑥 must be greater than negative 12. It’s not equal to negative 12 but is greater than negative 12. Here’s what 𝑥 can be, the parts that are highlighted in pink.
In set notation if a value can be greater than but not equal to, we use a bracket facing outward and then we use a bracket facing inward if it’s equal to. The range of 𝑥 is greater than negative 12 and less than or equal to negative four. Here’s one form of the interval notation. We can also note that in some places they use a curved open parenthesis when it’s not equal to. And sometimes they use the bracket facing outward. Both of them are describing the same thing. Both of them are the interval notation for this inequality.