# How to write an inequality equation from two boundary inequalities of a variable

I couldn't solve an SAT math problem (farmer picking pumpkins of the right weight and asking what ranges will he NOT pick) where I manipulated the word problem on a number line graph to give  x<2 and x>10.  I was asked to pick the answer that could be the correct one.

The answers I had to choose from were in the form of absolute value inequality equations, I solved all five answers and found that the answer (D)  |x-6| >4  was the correct answer.  This is one way to do this, grunt work/crank it out, but I want to explain to my student (AND MYSELF) how to do it analytically.

It appears that what I want to do is generate an absolute value inequality equation from data, seems simple enough, but I cannot find any references on the internet where this is done, it's always the other way.

Can someone explain to me the logic of how to do it?

From looking at the steps I went through to solve the answer and thinking going the other direction, the first thing I saw was rearranging both answers so both inequalities are in the same direction so on a number line I have an interval of a width[ (-2 to 10: 12), not (2 to 10: 8) ] and then I saw the center-point of the interval (4) was the same as in (D), and then I thought rely on the symmetry of the plotted absolute value function with the vertex at the midpoint, then taking the two "same direction of inequality" answers ( -x>-2 and x>10) and forcing them to both be equivalent to the mid-point value, and that worked, so I see what to do, but I don't understand the underlying math principles and the "why" of the "what" to do.

Why do I need a new interval with inequalities in the same direction?  Why is the midpoint of the new interval also the RHS value of answers?  Is this the key and the "why"?

PS--any internet references you can refer me to, or what Google terms to use to get them to show up?

Please go the definition of the absolute value: It is the distance of the a point from zero. So the inequality abs(x) > 4 can be read as abs(x -0) > 4, where 0 is the midpoint between -4 and 4, the solutions to the equation abs(x) = 4.

Now coming to your problem, the two inequalities are x < 2 and x > 10. The midpoint now is (2 + 100)/2 = 6, the origin shifts 6 units to the right, hence abs(x -6) and the distance of endpoint from midpoint is 4, hence abs(x - 6) > 4.

Hope this helps.

\$37p/h

Mike A.

Tutor college/high school/middle school-science/mathematics

2000+ hours
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