Hi! May you please explain how to solve number 4 algebraically from the practice questions from the “working with advanced systems of equations” chapter? Thanks!

Sure (although I recommend using a graph to visualize even if you insist on the algebra). Here it is for those following along:

First, recognize that in questions like this, the min or max value you seek will almost always be an intersection point. So we can kinda ignore the inequalities and just pretend they’re equal signs.

Because we know that y=5 when x=a, let’s substitute and solve for a.

5= (a-2)^2+4\\ 5= -(a-2)^2+6\\\\ (a - 2)^2+4=-(a-2)^2+6\\ (a^2-4a+4)+4=-(a^2-4a+4)+6\\ a^2-4a+8=-a^2+4a+2\\ 2a^2-8a+6=0\\ a^2-4a+3=0\\ (a-3)(a-1)=0

So a can equal 3 or 1. Because we’re looking for the least possible value of a, the answer is 1.

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