How can I understand $a−c>b−d \iff a - b > c - d$ without algebra?

Question

As a high school senior, I know that $\color{limegreen}{a}−c >b−\color{skyblue}{d} \iff \color{limegreen}{a}−c \color{red}{+c - b} > \color{red}{+c - b} +b−\color{skyblue}{d} \iff \color{limegreen}{a} - b > c - \color{skyblue}{d}$.

But how can I understand this without rearranging algebra? This answer betrayed to me that I couldn't intuit it.

I tried with real numbers and drew a number line, but I still can't intuit $\color{limegreen}{9}−3 >5−\color{skyblue}{2} \iff \color{limegreen}{9} - 5 > 3 - \color{skyblue}{2}$.

Answer 1

Suppose my height is $a-c$ and I stand with my feet at height $c$ and my head at height $a$, and you do the same with the numbers $b,\,d$ in place of $a,\,c$. So the statement is that I'm taller than you if and only if my head is higher compared to yours than my feet are to yours.

Answer 2

Note that $x>y$ is defined as $ x-y>0$ Thus

$$a−c>b−d \iff (a-c)-(b-d)>0 \\\iff a-c-b+d>0 \iff (a-b)-(c-d)>0 \\\iff a - b > c - d$$

Answer 3

You start with the task of comparing group sizes; and on assumption that in group A there $(a-c)$ people and in group B there are $(b-d)$ people. You assume that A has more people than B and then realize that if you add or discount any number from both, say $(c-b)$ people, you should still be left in a situation where A has more people from B. Also, assuming we don't know which group size was bigger but that after adding the same constant to both we got that A was larger than B, then originally A was larger than B as well. At last you notice that $(a - c) + (c - b) =a - b$ and so on.

Answer 4

"But how can I understand this without rearranging algebra? "

Why should anyone be able to do that? I can't do that. I never assumed anyone else could.

$9 - 3 > 5 - 2 \iff $

$9 \color{red}{- 5} - 3 > 5 \color{red}{- 5} -2 \iff $

$9 \color{red}{- 5} - 3 > 0 - 2 \iff$

$9 \color{red}{- 5} - 3 \color{blue}{+3} > 0\color{blue}{+3} -2 \iff$

$9 \color{red}{- 5} > \color{blue}{3} -2$

The thing is you get very used to doing it quickly. You can "intuit" manipulating and (in)equality be "swinging" the opposites to the other side. It's just like tossing a monkey to your sister while riding a bike and surely everyone has done that.

$ a + b = c \iff a = c - b \iff a-c = -b \iff -c = - b - a$... just ... swing those numbers around!

$a - b > c - d \iff d + a - b > c \iff d-b > c - a$. Just toss those monkeys! It's just like dancing, isn't it?

Now toss two monkeys at once:

$a - b > c- d \iff d-b > c- a$.