Video Transcript
Use the graph to find the solution set of the inequality 𝑓 of 𝑥 is greater than or equal to 𝑔 of 𝑥.
We’ll begin by defining our individual functions. We can see from the graph that 𝑓 of 𝑥 is the absolute value of 𝑥 plus one. 𝑔 of 𝑥 is given by this horizontal line. 𝑔 of 𝑥 is equal to three. So we’re going to use the graph to find the solution set of the inequality the absolute value of 𝑥 plus one is greater than or equal to three. And so, really, to solve this, we’re going to find the values of 𝑥 such that the graph of 𝑓 of 𝑥 is greater than the graph of 𝑔 of 𝑥.
Well, we can see that that’s in these two places. In fact, since we’re working with a weak inequality — that is, the absolute value of 𝑥 plus one is greater than or equal to three — we include the points where the graphs intersect. So we can say that one range of solutions are 𝑥-values that are greater than or equal to two. And the other range of solutions are 𝑥-values less than or equal to negative four. But remember, we’re looking to find a solution set. So how do we represent this using set notation?
Well, we’re going to consider the inverse of what we just stated. We know that the values of 𝑥 that don’t satisfy our inequality are the values of 𝑥 from negative four to two, but not including negative four and two. So that’s the open interval from negative four to two. And so the solution set of our inequality is the set of all real numbers minus the set of numbers in this open interval. So that’s the set of all real numbers minus those in the open interval from negative four to two.
