Test 1 Section 4 #28!

Now here’s a question I love because it’s got a really easy solution if you take a step back from it. You’re asked about the system of inequalities y\geq 2x+1 and y>\frac{1}{2}x-1. Solutions to a system of inequalities are the overlap of both individual solution regions, so really all you need to do to solve this is to recognize that the first inequality will have no solutions in Quadrant IV: it has a positive slope and a positive y-intercept.

If the first inequality has no solutions in Quadrant IV, then the system can have no solutions in Quadrant IV.

Below is an illustration of the full solution, which you can draw a less pretty version of based on the slopes and intercepts given. The blue region is the solution to the first inequality, and the red region is the solution to the second inequality. The overlapping (purple) region is the solution to the system of the two inequalities. See how the blue region never touches Quadrant IV? That’s why I’m saying that’s the only inequality you need to pay attention to if it’s the first one you think about.


Comments (2)

Hi! Just a quick question – how does an inequlity having a positive slope/positive y-intercept help determine that it wont contain any points from quadrant IV?

A line with a positive slope and positive y-intercept will never enter quadrant IV, so if the inequality is > or ≥, which means all points above the line, then none of those points will be in quadrant IV. See the blue line and blue shading in the picture above.

Leave a Reply