Baby Algebra Upgrades: Inequalities and Absolute Values

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Welcome back to the Ivy Lounge Test Prep series on upgrading your baby Algebra skills! In this series, I’m 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.)

So, 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. 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?

Inequalities & Absolute Values!

On the surface, they seem easy enough. But as we’ll soon see, you’re expected to have a much more sophisticated understanding of them than you did in 6th grade! So let’s start with the basics.


Inequality Basics

An inequality is like an equation, only instead of an equal sign, you have 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 ≥.

What you do to one side, you do to the other, just like in an equation. If there are THREE sides, like 3 ≤ 2x + 4 < 20 , then you’d do the same thing to all THREE sides.

The only little snag? When you multiply or divide by a negative number, you reverse the direction of the inequality arrows!

For example, 3 ≤ −2x becomes −3/2 ≥ x (because I had to divide both sides by -2).

Absolute Value Basics

Absolute Values are like parentheses with superpowers. For the sake of PEMDAS (“order of operations”), you must complete whatever’s inside them first. Only THEN can the super power come out!

Ex: | 2 – 12*3 | = | 2 – 36 | = | -34 |

Now what’s this “super power” I speak of? Simply that, 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’s refreshed, 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!

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Baby Algebra Upgrade #5: “The Distance/Difference Between”

I wish this one came up more often when people are learning to translate English into math, but it doesn’t, so I’m putting it here.

If I want to know 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 |

Crazy, right?

So “the difference between the real score (s) and her projected score (p) is 5” magically becomes

| s – p | = 5 OR | p – s | = 5

Or “the distance between her house (h) and her studio (s) is less than 10 miles” becomes

| h – s | < 10 OR | s – h | < 10

See how that closes the gap between English and math?

Once you can put a word problem into equation form successfully, you’re most of the way to solving it on an SAT math question!

So that’s it for now! You’ve successfully unearthed Inequalities and Absolute Values from the little attic in your brain and we’ve blown off the dust! In addition, you now have a handy-dandy NEW trick up your sleeve that you’ve most likely never heard before, which will make word problems just THAT much easier.

The next—and final—installment of our “Baby Algebra Upgrade” series is up next. It’s for sure something you have NOT learned in school—but that often shows up on the SAT and ACT. Stay tuned!