# At a Glance - Written Inequalities

Some word problems lead us to inequalities rather than equations, which is good, because algebra would be awfully boring if we never got to mix it up. We know what you're thinking, but don't say it.

When we're solving an inequality to answer a word problem, we need to think about the answer before we write it down, especially for problems where answers need to be integers.

### Sample Problem

Liam has \$20. He wants to buy some comic books that cost \$1.75 each. He doesn't like comic books, but his mom is forcing him to start a hobby so he doesn't spend all day, every day watching television. How many comic books can he afford to buy?

Liam only has \$20, so we need this inequality to be true:

(1.75)(the number of comic books Liam buys) ≤ 20

The above is true because we're assuming Liam won't be able to sweet-talk his way into any additional comic books, and that he won't go back into his wallet for a credit card. Probably a safe assumption to make.

Let x be the number of comic books Liam buys, and translate the inequality into symbols as:

1.75x ≤ 20

Solving this inequality gives us some horrible non-integer between 11 and 12. It's approximately 11.4, but actually a number that causes much more of a headache to look at. Liam can't buy 12 comic books, because he doesn't have enough money. He also can't buy part of a comic book, as they usually won't let you tear out a bunch of individual pages, so he can buy 11 whole comic books. Plus, he'll still have a few cents leftover to throw into a fountain and wish he were back at home watching TV.

#### Exercise 1

A rectangle has dimensions x by 2x. What is x if the perimeter of the rectangle is 48? Would it make this problem seem more relatable if we said it was a candy bar? Okay, it's a candy bar. Boom.

#### Exercise 2

The average of two numbers is 6. One number is 5 bigger than the other. What are the two numbers?

#### Exercise 3

Dumbo the elephant likes to eat peanuts. What elephant doesn't? He eats all the peanuts the visitors to the zoo bring him. He made a New Year's Resolution to cut back, but they're always right there. If the visitors bring 100 peanuts and Dumbo eats them all over the space of four hours, how many peanuts per hour does Dumbo eat? Bonus question: how much liquid Maalox will he need to counteract the acid reflux brought on by his excessive peanut intake?

#### Exercise 4

Esmeralda wants to buy some exotic fish with blue speckles and long flowing fins. Mmm...delicious. The person at the fish store told Esmeralda that these particular fish need a lot of water in their tank in order for them to be comfortable: 1 gallon for the tank to start, plus an additional 2 gallons for each fish she throws in there. Esmeralda has a tank that holds 64 gallons of water. How many fish can Esmeralda's tank comfortably hold? Let's assume that Esmeralda does not actually intend to eat these guys, and therefore does have some amount of concern for their comfort. She's not an animal.