Question Video: Graphing Systems of Inequalities with Parabolas

Video Transcript

Determine which region in the graph contains all of the solutions to the following system of inequalities: 𝑦 is greater than or equal to four times the absolute value of 𝑥 minus two plus one-half, 𝑦 is less than negative one-fourth times 𝑥 minus three squared plus four, and 𝑦 is less than 1.5.

So first, let’s notice that some of these functions have dashed lines, while others have solid lines. If the inequality has a less than or a greater than sign, it should be a dashed line. If it’s a less than or equal to or greater than or equal to, it should be a solid line. And as for shading, if it’s a less than or less than or equal to symbol, we shade below. And if it’s a greater than or greater than or equal to symbol, we shade above.

So we have three different inequalities: one is an absolute value graph, another is a parabola, a quadratic, and then the last one is linear — it’s a straight line. Absolute value graphs should be shaped like a V. Quadratic should be shaped like a U. And here is a linear one.

So right away, we can kind of tell which graphs go with which inequality. The absolute value also has a greater than or equal to symbol, meaning it should have a solid line and we should shade above it, which is here. So the blue is the shading above that. And then our quadratic should be shaped like a U, except our U is upside down. That’s because of this negative. So instead, it’s shaped like this. And since it’s a less than symbol, we will have a dashed line and it will be below, which we have in red. That’s the shading for it. We’re shading below that dashed line.

And then, lastly, 𝑦 is less than 1.5. 𝑦 is equal to 1.5 here at this line and it’s a dashed line since it’s a less than symbol. And we would want to shade below that, which we can kind of see in the graph. So let’s begin by kind of shading each one again. Beginning with the absolute value, we were supposed to shade above because it’s a greater than or equal to, which we have right here in the blue lines.

Next, we have the quadratic and it’s upside down and we shade below it, which is in the red, which we’ve now used with the blue lines. So wherever all three overlap will be our final answer. So right now, they overlap inside of the V. And then, lastly, we have the line 𝑦 is less than 1.5. So we shade below it. So we can see that here in the blue lines. So where do all three of these overlap? It would be right here in this tiny blue triangle, which would be region G.

So the region G will be our final answer.