I've seen identities, which means they have infinite solutions.$\longleftarrow$This is incorrect; see the comments below. Examples are$$4x+6=2(2x+3)$$$$9q-6\lt9q+3$$$$12u\le3(4u)$$$${x\over 9}={x\over 3\cdot3}$$$$15a\ge15a-2$$The first example evaluates to $4x+6=4x+6$ using the Distributive Property. How is it possible that (maybe some) of these equations or inequalities are identities? Maybe they have the same variable terms on both sides of each example and the constants are the same in an equation and one constant is bigger/smaller than the other constant on the other side depending on the inequality symbol. Maybe that's what makes them identities! I want to hear from your comfortable answers!
[Math] How are some equations and inequalities called identities (how do they have infinite solutions?)
inequality
Best Answer
To make your own identity, start with something that you know is definitely true, such as: $$ 1 + 1 = 2 $$ Now make the identity more complicated by doing legal (and reversible) operations to both sides of the equation. For example, we can add $x$ to both sides: $$ 1 + 1 + x = 2 + x $$ then multiply both sides by $3$: $$ 3(1 + 1 + x) = 6 + 3x $$ and so on. Notice that at any point, we may reverse our steps to get back to our initial identity.