Closed 2 years ago.
You are watching: When to flip signs in inequalities
Is over there a mathematical reason (like a proof) the why this happens? You can do it with examples and it is "intuitive." however the proof of why this happens is never presented in pedagogy, we simply warn students come remember to upper and lower reversal the inequality whenmultiply or division by a an unfavorable number both sides
$$-2>-3 \implies 2take reciprocals of exact same sign fountain both sides
$$\frac34 > \frac12 \implies \frac43
I"m slightly pertained to that
Is over there a mathematical factor (like a proof) that why this happens?
is a completely mathematical question, but since you create "we simply warn students" I will certainly assume the this inquiry is purposefully asked below on mathematics Educators sdrta.net.
As to a proof:
Given $a>b$, subtract $a$ native both sides: $0 > b-a$.
Next, subtract $b$ from both sides: $-b > -a$.
Note the this final inequality is indistinguishable to $-a .
And so we have actually proved: If $a > b$, then $-a .
Depending top top the context and also the ahead curriculum, the following can work:"less than" method "to the left of" ~ above the number line. Multiplying by a an adverse number flips numbers approximately 0.Thus, "left of" i do not care "right of", or "greater than".
Perhaps one means to check out (and define intuitively to children) the "multiply by $-1$ part" is the following. Imagine your 2 numbers, $a$ and also $b$, lied on the numberline. Multiply by $-1$ is prefer "rotating the numberline through 180°": imagine it"s a straight metal pole, lying flat on the ground; pick it up by the middle, and rotate the "long ways" (ie no "barrel roll") 180°. (One only demands to consider the line segment $<-\max\a,b\,\max\a,b\>$ because that this, in case you acquire a smart-arse saying you can"t relocate something of limitless mass!)It is hopefully clean to most human being why the point that was to the best is currently to the left.
See more: How To Say I Love You In Punjabi ? How Do You Say 'I Love You' In Punjabi
This is only a intuitive intuition -- one of course needs to do this a rigorous proof, and also the various other answers perform this -- but such intuition can frequently be an useful to early-learners.
A similar, yet slightly less clean, statement deserve to be made about inversion: take just the optimistic real axis $(0,\infty)$; turn it about the number $1$. (Since, again, we have the right to look in ~ $(0,M)$ because that say $M = a + b$ (with $a,b > 0$), one could imagine some sort of ellipse.)