Compound Inequalities

Question

-6x+14 < -28 3x +28 <_ 25

x > 7 x <_ -1

I figured the answer was: x >7 OR x <_ -1

How do my answers conclude with NO solutions?

Answer 1

Hi Brian P.,

Your error happened in the assumption of "OR" between the two individual inequality solutions. The overall solution must satisfy both inequalities, and the "AND" condition between those two sets is {}, or the null set == no solutions.

Disclaimer! math has traditionally been approached as relationships between numbers, e.g. numbers which satisfy various conditions. A more powerful, but much more involved way of 'doing math' is peeping into sight recently, viewing it as relationships between categories. Everything that was true between numbers, could be translated into statements about categories, but much more can be examined, too. Note that this is NOT the same as the usual way of saying that sets are equal or identical if they have exactly the same set of elements, [which could be numbers, could be operations (such as rotations, reflections, inversions, etc. on a plane or in higher-D spaces), could be various other sorts of things too] -- it is a fundamental change that gets rid of the notion of solving equalities per se. Reference:

https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/

Note: This is heady stuff, but don't be scared off by things that you don't understand; this is a considerately-written overview. AND it's an area of math that's still wide open to discovery!

-- Cheers, -- Mr. d.

Answer 2

"-6x+14 < -28 3x +28 <_ 25"

I see this as 2 inequalities:

-6x+14 < -28

which simplifies to:

-3x + 7 < -28

-3x < -35

x > 35/3

3x +28 ≤ 25

which simplifies to:

3x ≤ -3

x ≤ -1

Answer 3

This type of compound inequality is an AND, not an OR. AND means that BOTH equations need to be true. Since your answer shows no overlap at all, there is no region where both are true. That is why the answer would be NO Solution.