>

## Linear Inequalities

I didn’t have to teach them this year. But I worked on a lesson for them with some other teachers.

I take it back – here’s a real world application:

Go to vegas.com. You’ll see tables that look like this:

| May. 7 |\3. Utah | | 7:35 p.m. |\3. at LA Lakers|
{background:#ddd}.| Favorite | LAL | LAL | LAL |
{background:#aaa}.| Point spread | -7 | -7 | -6½ |
{background:#ddd}.| Over/Under | 209 1/2 | 209 1/2 | 209 1/2 |
{background:#aaa}.| Total money line | -110 | -110 | -110 |

You can discuss all the possible scores that’ll let you win one way, or the other.

You can do it for both the spread (L-7 <> J), and the O/U (L + J <> 209 1/2).

And then you can discuss what happens when you want to bet on both at once:

Wahla! Systems of inequalities!

1. Clint H | May 6, 2008 at 7:13 am | Permalink

Whoa. Math, gambling, and the Lakers? In one post? And no probability? Sweet. Now I just have to explain O/U and the line to my 8th graders…

2. Jason Dyer | May 6, 2008 at 2:32 pm | Permalink

I admit I’m totally understanding — could you explain a little more in detail how this works?

3. Jason Dyer | May 6, 2008 at 2:33 pm | Permalink

NOT understanding, of course. Typo city.

4. Mr. K | May 6, 2008 at 6:11 pm | Permalink

Sure.

(I’m going to use L to represent the number of points the lakers score, and J to represent the number of points the Jazz score).

When you talk to your bookie, there are a number of ways you can bet. A lot of them involve something that the bookie thinks is a 50/50 chance, so that when you win you basically double your money, minus a small commission for the service.

One of these bets is a point spread, where they try to predict by how many points a team will win. (Or, really, in economic terms, they try to find the point where people will bet an equal amount of money on either side).

In the example above, the spread is Lakers -7. You can bet on the Lakers to cover the spread, which would mean them having to win by more than 7 points. Or you could bet on Utah to cover, and win the bet even if the Lakers win, as long as they win by less than 7. If I was betting on the Lakers, this would be in the dark blue or gold section of the graph up there – everything higher than the line of L = J + 7. Betting on Utah would be below that line.

Another type of bet is the over/under: You bet on whether the total of both teams scores will be greater or smaller than some number. The bookie uses the same logic to pick that value: they want the same amounts bet on both sides. However, it doesn’t matter who wins. It just matters if there is a lot of scoring, or not a lot. The dark blue & purple areas represent the under, and the teal & gold are the over. The line separating those two areas is described by L + J = 209.5

For some people, doubling their money isn’t enough. As you can see above, the two bets are orthogonal – you can vary one without affecting the other, so a lot of people will place both kinds of bets.

So, lets say we want to bet for the Lakers to cover the spread. That’s represented by the inequality L – 7 > J. Graphically, that’s everything above the upwards slanted line.

Lets say that, since everyone plays tougher defense in the playoffs, we also expect a lower scoring game, so we want to bet the under. That’s represented by the inequality L + J < 209.5 That’s everything below the downward slanting line.

In order to win both bets, we need to be both below the downward slanting line, and above the upwards slanting line. On the graph, this is represented by the dark blue area, and corresponds to the system of inequalities for L – 7 > J and L + J < 209.5

Switch the bets, and you switch to one of the other corresponding areas to win big.