# Equations and Inequalities

### Topics

## Introduction to :

### How Many Solutions Can an Equation Have?

Okay, algebristas,* different equations can have different numbers of solutions. Right now, we are only counting solutions from the real numbers. Some equations have solutions that are imaginary numbers, but we'll get to those later. No, you don't need to send us a Thank You card. Your words are enough.

*You know, people who serve mathuccinos. Alternatively, you.

### Sample Problems

- The equation
*x*= 5 has only one solution: the number 5. - The equation
*z*^{2}= 4 has two solutions:*z*= 2 and*z*= - 2. - The equation
*x*=*x*has infinitely many solutions: any value of*x*will work, since*x*is always equal to itself. - The equation
*y*^{2}= - 5 has no real number solutions because the square of any real number is positive.

We interrupt this program to bring you a History Snack. Don't let it ruin your appetite.

Diophantus was a famous mathematician who's sometimes called "The Father of Algebra." Although with the way he got around, who *wasn't* he the father of? #ancientgossip

Anyway, D-man only liked positive rational solutions to equations. He would probably call a lot of the equations we will be solving "absurd" since they have negative solutions. It's hard to blame him, though—after all, there's no such thing as a negative child. (He wishes. Oh, burn.)

#### Exercise 1

Figure out how many solutions there are to the equation *x* = *x* + 1.

#### Exercise 2

Figure out how many solutions there are to the equation *w* + 3 = 7.

#### Exercise 3

Figure out how many solutions there are to the equation 2*x* = *x*.

#### Exercise 4

Figure out how many solutions there are to the equation *x*^{4} = 1.