Hello there, and welcome back to the Ivy Lounge Test Prep® series on leveling up your baby Algebra skills! In this series, I’m helping you revisit the math skills that you learned back in middle school….so that you can snag the (many!) points that the SAT and ACT offer you for applying these apparently simple skills in sophisticated ways. (If you didn’t catch the first post of this series, which explains why it’s smart to return to these fundamentals as you study for the SAT/ACT, you can read that here.)
You’ll also want to make sure you check out the previous Baby Algebra topics I’ve written about: the basics of linear equations; the fact that systems of linear equations crop up in more places/forms than you might have thought; and how to recognize how many solutions a system of linear equations has. That’s a lot of upgrading! But do you know what other “baby algebra” concepts come up on the SAT and ACT that you probably haven’t visited since elementary school? Well, you probably guessed it from the post’s title:
Inequalities & Absolute Values
At first glance, they seem simple enough. But as we’ll soon see, the ACT and SAT demand that you have a much more sophisticated understanding of them than you did in middle school! So let’s start with the basics.
Inequality Basics for the SAT and ACT
An inequality is a lot like an equation—except there’s no equal sign, but rather, signs that give you ranges of values. Specifically, we have “less than,” “greater than,” “less than or equal to,” and “greater than or equal to”: < , > , ≤ , and ≥.
As with an equation, with inequalities, what you do to one side, you must also do to the other. If there are THREE sides in an inequality (as has been known to happen), like 3 ≤ 2x + 4 < 20 , then you’d do the same thing to all THREE sides.
The tricky part? If you divide or multiply by a negative number, you reverse the direction of the inequality arrows!
For example, 5 ≤ −2x becomes −5/2 ≥ x (because I had to divide both sides by -2).
Absolute Value Fundamentals on the ACT and SAT
Absolute Values are like parentheses with secret powers. For the sake of PEMDAS (“order of operations”), you must complete whatever’s inside them first. Only THEN can the magic power come out!
Ex: | 2 – 12*3 | = | 2 – 36 | = | -34 |
And what about this “magic power” I mentioned? Well, AFTER simplifying whatever’s inside first, if the value I’m left with on the inside of the absolute value lines is negative, I get to “turn that frown upside down”... you know, make it positive!
Ex: | -34 | = 34 :)
So, now that that we’ve reviewed that subject from your tender middle school years, let’s “upgrade” the baby Algebra! This time, with concepts that use both inequalities AND absolute values together! We’ll start with a concept that helps you understand how to apply these skills—and then next week, I’ll teach you something you may never have seen before!
What Do Standardized Tests Mean by “the Distance/Difference Between”?
This is such a handy, but under-discussed, topic when it comes to translating English into math.
If I need to figure out the “distance between”—or even the “difference between”—two variables a and b, here’s how I write that out:
| a – b | OR | b – a |
Wacky, right?
So “the difference between the real score (r) and her projected score (p) is 6” magically becomes
| r – p | = 6 OR | p – r | = 6
Or “the distance between her apartment (a) and her workplace (w) is less than 9 miles” becomes
| a – w | < 9 OR | w – a | < 9
Isn’t it handy to understand how English translates into math?
When you know how to transform a word problem into its equation form, you’re close to solving it on the SAT Math!
So that’s it for now! You’ve successfully dredged up Inequalities and Absolute Values from the storage unit of your deep memory, and we’ve shined it up ‘til it looks new again! In addition, you’ve gained a very helpful NEW shortcut that you’ve most likely never encountered before—a shortcut that will make word problems just THAT much easier.
The next—and final—installment of our “Baby Algebra Upgrade” series is up next. There’s a 99% chance you didn’t learn about it in school—but it often shows up on the SAT and ACT. Sounds like something you need a blog post on! (And, pssssst, if you’d like way more math hacks in this vein, check out my Math Cram Plan ebooks for the SAT and for the ACT.)
