Why does $x= (5/3)$ behave the same as $x=(-5/-3)$ in the equation $2x + 3 = 5x - 2$?

Question

Solving for $x$ in the original equation $(2x + 3 = 5x - 2)$ I got $-5/-3$, but the video I was watching from got the value $5/3$. Checking the correctness, I substituted my answer and got $0$ on both ends, meaning that the two expressions were equal. Can someone please explain why the negative gives the same result as the positive?

Answer 1

The negatives cancel out. This is the same thing as saying $$\require{cancel} \frac{\cancel{(-1)}(5)}{\cancel{(-1)}(3)}$$

Remember, a negative times a negative is positive, and likewise, a negative divided by a negative is positive.

Answer 2

A debt of \$5 is almost twice as much as a debt of \$3, just as a deposit of \$5 is almost twice as much as a deposit of \$3. The ratios are the same.

Answer 3

Two rational numbers $a/b$ and $c/d$ ($a,b,c,d$ integers, $b\ne0$ and $d\ne0$) are equal if and only if $$ ad=bc $$ By definition, I should add.