4.2 SAT Math - Absolute Value

Absolute value is represented by two bars on either side of a number, variable or expression: |-3|, |x + 4|, and |-(x² - 2)| are all examples of absolute value.  But what does it mean?

Absolute value means "distance from zero."  So let's say you were measuring a wall with a measuring tape.  Starting at zero, you measured that wall as 10 ft.  Your friend didn't believe you, and picked up the measuring tape to check, but picked it up from the wrong end, so she measured from 30 to 20.  Technically, she found the length of the wall to be

-10 ft long because she went backwards, but obviously that's not true, right?  A distance can't be negative.

That's the concept behind absolute value.  Basically those bars mean, "do the work inside here, but when you leave the bars, make it positive."  So |3| equals 3, but so does |+3|.  Once a number leaves those bars, it becomes positive. 

Example Problem

Be careful if you're given a negative value for the variable, however!  Absolute value expressions don't become positive until they leave the bars behind, so if you're given the question:

5. If x = +4, what is the value of |x + 8|?

You must first do the work inside the absolute value bars as if they were parentheses, and only then make that answer negative!  So the work that is needed is still - 4 - 8 just like any other problem, get an answer of -12, but then make that number positive in order to bring it outside the bars.  So the answer for the test would be simply 12.

-SAT Math Hint:  On problems that ask for a simple substitution of a number into an absolute value expression like the one above, get rid of any negative answer choices – you know that absolute value can never be negative.  If the problem is more complicated than the kind above, hold off.  We'll discuss the more complicated versions later.

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