5.3 SAT Math - Absolute Value Equations and Inequalities

We've gone over the basics of absolute value for the SAT, but when you put them into equations or inequalities you get something a little harder. 

You remember that absolute value means the equation inside the bars could be positive or negative.   When you're simply substituting a number in for your variable, like "what is the value of |x + 5| if x=-4," you simply solve inside the bars, and then make that number positive.  But what about when you're given an equation, say |3 + y| = 6?  "Easy," you might say, "make it positive."  Not so fast.  The problem is, the stuff inside the bars could be positive or negative.  We don't know, and they don't tell us.  So you have to solve it two different ways, once in case it's positive, and once in case it's negative, in order to have both possible correct answers.  The way to do that is to simply set up exactly those two equations: 3 + y = 6 and 3 + y = -6.  Solve for y in each equation and you will have your two possible answers:

3 + y = 6 and 3 + y = -6

y = 3 and y = -9

If you plug both of those back into the original equation, you'll get |6| and |-6|, which we know works out to positive 6.  So the answer is y = 3 or y = -9.

These can get a little trickier when they give you more 'junk' around the absolute value bars.
-SAT Math Hint: In order to solve any absolute value equations or inequalities, you have to have the absolute value part on one side and everything else on the other BEFORE you split the equation into two equations.

So if you're given: 5+ 2|x - 6| = 13, you first need to clean that up to get the absolute value part all by itself.  Clean it up as if the absolute value bars were parentheses you wanted to keep:

5 + 2|x - 6| = 13

2 |x - 6| = 8

|x - 6| = 4

From there, you can split it into equations and solve for your two possible values of x.

Inequalities

Inequalities get even trickier, but not much.  Inequalities follow the same rules (get the absolute value on one side and everything else on the other, split it into two equations, one positive and one negative), but because of the inequality rule regarding multiplying/dividing by negatives, we don't just set the second equation equal to a negative, you also must flip the sign.  Take a look:

12. If |3x -4| + 6 ≤ 10, what are the values of x?

First, clean up the original equation to get the absolute value by itself. 

|3x - 4| + 6 ≤ 10

|3x - 4| ≤ 4

From there, setting up the first equation is easy.  Simply rewrite the equation without the absolute value bars: 3x - 4 ≤ 4.  To set up the second equation, change the number side to negative, and flip your sign: 3x - 4 ≥ -4.  Now, solve each:

3x - 4 ≤ 4 and 3x - 4 ≥ -4
3x ≤ 8 and 3x ≥ 0
x ≤ 8/3 and 3≥ 0

Now the question remains, is this an "and" question, or an "or" question?  What that means is, if you were to graph this on a number line, would the arrows be going towards each other or away?  Does x equal only the numbers between these two, or are the numbers between these two the only ones excluded?

You can always graph it on a number line to be sure, and some questions may even require you to do so, but there's a great shortcut you can use:
-SAT Math Hint: Look to the original equation.  If the original equation had a less than sign (<) it's an "and".  If the original equation had a greater than sign (>) it's an "or."  How can you remember that?  Think "less thand" and "greator."

That means in this equation, x ≤ 8/3 and 3≥ 0.  "And" answers can be combined: 0 ≤ x ≤ 8/3 and usually will be for the answer choices.  Just make sure the arrow is pointing to the number you want it to point to when you go to the answers, and you should be fine.