Week 2 Completed
I am writing this post on 15th October 2023. I have completed the second week of Term 1 at university. Things are mostly going ok and I am beginning to get into the flow of work-life balance. Again, this is a free post on a challenging area and algebra problem.
We want to manipulate the first expression into simpler forms so that we can identify the area on the xy-plane.
The paid newsletter contains more abstract university-level maths in bite-sized form. Last week, we covered pointwise convergence, an extension of the study of limits from high school calculus.
Anyway, let’s give this problem a try before jumping in for the solution!
Solution
While the first inequality given looks somewhat daunting, we can use the technique of factorization. Our goal is to break it down into a product of expressions so that we can understand the inequality.
We first take out the common factor of each term.
We then group the terms inside the bracket into the following. It should be seen that -(x - y) is the same as y - x.
This step uses the difference of two squares. We can now see that this can be nicely factorized into the following expression.
Now that we have the factorized expression, let’s try and understand how the second constraint impacts the first one.
The second condition states that x is less than or equal to y, this tells us the two following inequalities must be true.
Notice the first condition is a product of three factors. We want the overall expression to be greater than or equal to zero. And given what we have deduced above, it must be true that x + y - 1 is less than or equal to zero.
It should be evident that the above is the only possible combination given our inequalities.
Therefore, the region of the xy-plane is defined by the following.
The first inequality simply tells us that the region must be in the 1st quadrant.
As for the other two inequalities. We can first plot the equations on the plane.
The intersection can be found by solving the two equations. (0, 1) is the y-intercept of the blue line.
So what is the region? From the inequalities, we can make y the subject for a better analysis.
The first inequality clearly shows that the region above the red line; the second inequality indicates the region below the blue line.
The region that satisfies both conditions is the one we want.
So what’s the area? We know it’s a triangle with a base of 1 and a height of 1/2.
Therefore, we have the following as our answer.
The answer is A.
Well done.
What was your approach?
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Happy reading,
Barry 🍩