# Applications with Inequalities

## Key Questions

• Examples

$x$ is at least $2$. $\iff$ $x \ge 2$

$x$ is no more than $5$. $\iff$ $x \le 5$

I hope that this was helpful.

• When we need to make some calculated number less than (or greater than, or not more than or not less than) some fixed number. Or when we need to made one calculated number less (or whatever) than another calculated number.

For instance, Suppose I want to go into the tee-shirt printing business. If I know that it will cost me $140 for material plus $12 per shirt printed, and I have $600 that I can spend getting started, what is the maximum number of tee-shirts I can produce? This means solving: $12 x + 140 \le 600\$.

(OK, it's not the best possible example, but that's the kind of thing we need it for in algebra. There are other kinds of problems in calculus that involve increasing and decreasing quantities.)