Video Transcript
Given that 𝑥 belongs in the set of integers, which of the following number lines represents the solution set of three 𝑥 minus five is greater than or equal to four 𝑥 minus three?
Let’s work through the process of solving this inequality and then consider how its solution would be represented on a number line. The variable appears on both sides of the inequality. And there are also constant terms on each side. We’ll start by collecting the 𝑥-terms on the same side of the inequality.
As the coefficient of 𝑥 is greater on the right-hand side, we’ll collect the 𝑥’s here so that we obtain a positive coefficient. Subtracting three 𝑥 from both sides of the inequality gives negative five is greater than or equal to 𝑥 minus three. Next, we’ll collect the constants on the left-hand side by adding three to each side to give negative two is greater than or equal to 𝑥.
It’s conventional to write the solution to an inequality with the variable on the left-hand side. So we can rewrite our solution in this way. We must be careful and make sure the inequality sign still points towards the 𝑥. So, instead of negative two is greater than or equal to 𝑥, we obtain the equivalent statement 𝑥 is less than or equal to negative two.
We’re almost ready to consider what the solution would look like when represented on a number line. But first we must remember that we were told in the question that 𝑥 is an integer. That means that 𝑥 can take any integer value less than or equal to negative two. So that’s negative two, negative three, negative four, negative five, and so on. This would be represented on a number line by dots above each individual value to indicate that the values are discrete, with an arrow pointing in the negative direction to indicate that all negative integers below these are also included.
Looking at the five options, this is what has been drawn in option (A). So option (A) represents the solution set of the given inequality. Notice that option (B) has dots drawn over the first four integers that are greater than or equal to negative two and an arrow pointing to the right. This number line would therefore represent the solution set of the inequality 𝑥 is greater than or equal to negative two in the set of integers.
Option (C) has a single dot over the value of negative two. So this would represent the solution 𝑥 equals negative two.
Option (D) is very similar to the correct answer. But the arrow indicating that all smaller integer values are also included has been omitted.
Option (E) is the discrete set of values negative two, negative one, zero, and positive one. So this could represent something like the solution set of the inequality 𝑥 is greater than or equal to negative two and less than or equal to one in the set of integers.
As we’ve already determined, the correct answer is (A).
