We’ve all seen how the GMAT loves to throw sneaky absolute values into data sufficiency questions involving number properties. Here’s a quick refresher on the properties of those double-bars, and a quick practice question!
The absolute value represents the distance from zero on a number line. Since a distance can never be negative, absolute values are always positive. On the GMAT, most absolute values you will see will involve a variable. Let’s consider |x|. If x is a positive number, such as 4, then |x| = x, because |4| = 4. However, if x is a negative number, such as negative 4, then |x| = -x. For any negative value of x, the sign would have to be changed.
For x > 0, |x| = x
For x < 0, |x| = -x.
To solve absolute value equations, we must split them into two separate equations, removing the absolute value, and making one equation negative. Let’s look at an example.
Notice how we solved two equations independently, getting two solutions for the absolute value. If you are confused, plug in x =4 and x = -6.5 back into the absolute value for x. Notice that they will both make the absolute value true.
Test Day Tip: Look out for absolute value questions that have words like “can” or “must” in them. Either solution is correct, but they don’t both have to be correct. x could equal 4 or it could equal -6.5, but it doesn’t have to only equal 4, and it doesn’t have to only equal -6.5. This is especially tested in Roman numeral questions.
Another way GMAC tries to complicate an absolute value question is by combining it with other topics, such as inequalities. Let’s see how an absolute value-inequality is solved:
Here we split the absolute value into two inequalities, as before, but notice how we FLIP THE SIGN of the inequality that becomes negative. That is because you must always flip the sign when you multiply or divide an inequality by a negative number. As we solved, we AGAIN flipped the sign when we divided by -3. This is an extremely important rule to remember, so watch out for those negatives when multiplying/dividing!
Here we can a range of possible solutions, rather than 2 solutions. Always graph your range on a number line to visualize the possible answers. Here we use open circles to show that x cannot actually equal either -2 or 4/3.
Here’s a tough lil’ guy from the GMATPrep:
(1) (x2) > (y2)
(2) x > y
Stumped? Check out this detailed explanation from Beat the GMAT, then dive into some more practice questions here at Veritas Prep!
Vivian Kerr is a regular contributor to the Veritas Prep blog, providing tips and tricks to help students better prepare for the GMAT and the SAT.