So, at long last, we’ve reached the end of my baby Algebra upgrades. In this series, I’ve been helping you level up the math skills that are supposed to be “easy” so that you can pick up the (many!) points that the SAT and ACT offer you for applying these apparently simple skills in sophisticated ways. (Check out this refresher course on why you may not know the math you think you know, if you missed the beginning of the series.)
We’ve done a lot of upgrading! After these posts, now you understand that more scenarios than you originally thought are actually systems of linear equations (and you now understand linear equations), and you also know how to recognize how many solutions a system of linear equations has. You’ve even refreshed your understanding of inequalities and absolute values. Whew! Give yourself a well-deserved pat on the back.
Now I’ve got one last trick for you—it’s one that will help you work quickly and capably with inequalities in word-problem formats on the SAT and let you crack into problems that might otherwise baffle you. And the best part? Although it’s simple and common-sense, my students tell me that it’s not taught in math classes—and I know I’ve never seen it in test-prep books. So this one’s my gift to you!
Baby Algebra Upgrade #6: Changing compound inequalities to single inequalities, and vice-versa.
Sometimes, you get a word problem with a possible range of values that is easily expressed as a compound inequality. For example, let’s say there’s a certain ride at an amusement park, for which you can ride if you’re at least 60” tall but no more than 78” tall.
Normally, I’d just write a compound inequality like this:
60" ≤ height ≤ 78"
However, the answer choices might not represent this range in the same way as you’re used to thinking of it. It might just represent this same information as one single inequality. This task for you in this scenario is to recognize the correct answer by being able to translate between the format you’re used to...and one you’re not.
If you find yourself in this conundrum, I’ll tell you how you crack it.
Here’s how you go from a compound inequality to a single inequality:
1) Find the average of the endpoints. In our example, I’d find the average of 60 and 78, which is 69, because (60+78)/2 = 138/2 = 69.
2) Find the “leeway”—as in, how far away is your average from either of your extreme values? In this example, our “leeway” is 9, because 69 is 9 away from 60 (the low extreme) and 69 is 9 away from 78 (the high extreme).
3) Put them together like this:
| variable – average | ≤ leeway
| average – variable | ≤ leeway
To round out our example, I could write either of the following and be correct:
| 69 – h | ≤ 9 OR | h – 69 | ≤ 9
And now let’s look at the reverse! Let’s say we are given the single inequality to represent that Amy wants her SAT score to be within 20 points of 1500. Or “the difference between her score and 1500 is less than 20.”
| 1500 – s | ≤ 20
To take a single inequality with absolute values and turn it into a compound inequality, all we do is this:
average – leeway ≤ variable ≤ average + leeway
In our test score example, it would be this:
1500 – 20 ≤ s ≤ 1500 + 20
which reduces to 1480 ≤ s ≤ 1520
And there you have it: a concept you’ll encounter in word problems rendered a solvable inequality. This kind of skill is important: it’s what keeps you moving quickly and capably through your math sections instead of sitting there stumped, losing time and confidence (…and the points that go with them!). All of my students work on these kind of baby Algebra upgrades…and it pays off for them in the scores they need to get into the schools they want. Now it can pay off for you too!
Consider your algebra SAT-ready!
This is literally the tip of the iceberg of upgrading simple Algebra skills, but these are major SAT point centers—even the kind of refreshing I can share with you in this kind of series will help a lot. If you’d like to upgrade your baby Algebra thoroughly and personally, you can contact me here.