Video: Finding the Solution Set of Linear Inequalities with Real Numbers

Video Transcript

Find the solution set of the inequality 𝑥 is less than or equal to 20 in the real numbers. Give your answer in interval notation.

Now, I said this automatically when reading the question, but the first thing to remember is that the inequality notation that’s been used here is a “less than or equal to” sign. So this inequality tells us that the value of 𝑥 is less than or equal to 20. This is known as a simple inequality as there is only one inequality sign involved. And actually, this inequality has already been solved for us as we have 𝑥 on its own on one side of the inequality. Picturing a number line, we know then that 𝑥 can take any value at all from 20 all the way down to negative ∞.

We can write this as an interval with negative ∞ and 20 as its endpoints. But we must be careful about the type of brackets or parentheses that we use at each end of the interval. As 𝑥 is less than or equal to 20, 20 is included in the possible values for 𝑥. So it is included in the solution set, which means our interval needs to be closed at its upper end. However, as negative ∞ does not exist as an actual number, our interval needs to be open at its lower end. We therefore have the solution set of this inequality in interval notation. It’s the interval from negative ∞ to 20, which is open at the lower end and closed at the upper end.