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Question 1003945: I am taking the GRE test and it has been a loooong time since my high school math. I am tripped up on how to simplify or transform one inequality to another. The book gives an example, but skips how to simplify it.
The question is as follows:
Which quantity is greater? x^2+1 or 2x-1
The book says that a quadratic polynomial can be factored by subtracting 2x from both sides
x^2-2x+1 ? -1
Then to factor the left hand side.
(x-1)^2 ? -1
I know a squared number has to be equal to or greater than zero so the left side is larger but I need more explanation of basic factoring.
Please get back to me at your earliest convenience. Thank you for your time.
Chris
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i'll give you some links on quadratic equations that will help you solve them.
the easiest way to solve this would be to graph it, assuming you are allowed to use a graphing calculator.
the graphing calculator would have showed you that x^2 + 1 is greater than 2x - 1 for all values of x.
that graph is shown below:
without a graphing calculator, you would need to do some analysis.
what they recommended is to get the quadratic on one side of the inequality and the constant term on the other.
your inequality is x^2 + 1 ? 2x - 1
the ? is there because you don't know what it will be just yet.
it could be =, >, <, >=, <=.
so just leave it at that to start with.
now subtract 2x from both sides of the inequality.
x^2 + 1 ? 2x - 1 becomes
x^2 - 2x + 1 ? -1
you now have the quadratic on the left and the constant on the right.
you could also have just moved everything to the left and you would have had:
x^2 - 2x + 2 ? 0
either way would have worked.
we'll stick with x^2 - 2x + 1 ? -1 for now.
rather than factor this, find the minimum point.
you know it has a minimum point because the coefficient of the x^2 term is positive.
if it were negative, you would have a maximum point.
the minminum point is at x = -b/2a when the equation is in standad form of ax^2 + bx + c = 0.
x^2 - 2x + 1 is in standard form when you set it equal to 0.
you get a = 1, b = -2, c = 1
x = -b/2a becomes x = 1
when x = 1, the equation of x^2 - 2x + 1 becomes 0.
your minimum point is at (1,0).
your inequality becomes:
0 ? -1
0 is greater than -1, so you get 0 > -1.
this translates to:
x^2 - 2x + 1 > -1
add 2x to both sides of this inequality to get x^2 + 1 > 2x - 1
you're done.
the graph os x^2 - 2x + 1 is shown below:
factoring was unnecessary in this case since what you were looking for was the minimum point on the quadratic and not the roots.
the followng references should help to refresh your memory about how to deal with quadratics.
i included a refresher on inequalities as well just in case you wanted to look at that as well.
http://www.purplemath.com/modules/solvquad.htm
http://www.regentsprep.org/regents/math/algebra/ac4/lpara.htm
http://www.mathsisfun.com/algebra/inequality-solving.html
there's lots more help on the web.
here's a couple of good websites to look at:
http://www.wtamu.edu/academic/anns/mps/math/mathlab/
http://www.regentsprep.org/
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