Inequalities and Absolute Values: Test the EDGES
Say you hit this on the Quant section: |x − 4| < 3. You plug in 5. It works. You feel sharp, you pick the matching answer, and you move on. Ninety seconds later the question is gone and so is the point. I watch people make this exact move every week. They are usually the strongest math people in the room.
Here is the blunt verdict: ONE number proves almost nothing on an inequality question. A friendly number like 2 or 5 will agree with almost any statement you throw at it. The trap lives at the EDGES: zero, negatives, fractions, and the exact boundary values. Test those and the trap shows itself. Skip them and the test takes your points without a fight.
The two words, in plain English
An inequality is a statement that one thing is bigger or smaller than another. x > 3 means x is more than 3. x ≤ 3 means x is 3 or less.
Absolute value, written with two bars like |x|, means distance from zero. Distance is never negative. So |5| = 5 and |−5| = 5. That second one is where people get hurt. Every absolute value statement has a positive side AND a negative side, and the people who write these questions know you will forget the negative one.
A cleaner way to read |x − 4| < 3 is: x is less than 3 steps away from 4. So x sits between 1 and 7, and it does not touch either end.
On the Executive Assessment, these show up in the Quantitative Reasoning section: 14 questions in 30 minutes, a mix of Problem Solving (regular five choice multiple choice) and Data Sufficiency, scored 0 to 20 and weighted the same as Verbal and Integrated Reasoning. You can see the full breakdown on GMAC's assessment structure page. Fourteen questions in 30 minutes is a little over 2 minutes each. You do not have time to guess and pray. You DO have time for 30 seconds of edge testing on paper.
The edge kit
When a question has an inequality or absolute value in it, these are the numbers worth testing:
- The boundary itself, where the statement flips from true to false
- 0
- 1 and −1
- A fraction between 0 and 1, like 1/2
- A negative fraction, like −1/2
- A big number, like 10 or 100
Why these? Because they behave strangely. Squaring 1/2 makes it SMALLER. Squaring −3 makes it positive. Multiplying by 0 wipes everything out. A plain 2 or 5 does none of that, which is exactly why it lies to you. Stop reading for a second and ask yourself: when you last missed an inequality question, which of these numbers did you skip? I'll bet you tested a number from the middle and never went near the ends.
The mistakes I see every week
Testing one friendly number and stopping. One number that works only shows the statement CAN be true. It does not show it is ALWAYS true.
INSTEAD: Test at least two numbers from the edge kit, and pick them on purpose to BREAK the statement. You are trying to prove it wrong. If you cannot, then you can trust it.
Forgetting the negative side of absolute value. |y| = 4 has two answers, 4 and −4. People find one and quit.
INSTEAD: Every time you see the bars, write two lines on your paper: the positive case and the negative case. Or write the sandwich, like we did above.
Multiplying or dividing by a negative without flipping the sign. −2x > 6 does NOT mean x > −3. It means x < −3.
INSTEAD: Flip the sign every time you multiply or divide by a negative, then check with a number. Try x = −4: −2 times −4 is 8, and 8 > 6. True. Try x = 0: 0 > 6 is false. Good, x < −3 holds up. And if you are about to divide by a variable and you do not know whether it is positive or negative, STOP. You cannot know whether to flip. Test values instead.
Treating < like ≤. One little line under the symbol decides whether the end number counts.
INSTEAD: Plug the boundary number in, on paper. One line of work settles it.
Doing all of this in your head. Some people think they are too good to write down x = −1/2. That is pride, and pride is how a strong math person ends up with a weak Quant score.
INSTEAD: Write each test value and its result in a little column. It takes seconds, and it is the same reason you should draw every geometry question: paper catches what your brain skips.
Where edges matter most: Data Sufficiency
Data Sufficiency gives you a question and two statements. You do not have to solve it. You only decide whether the information is ENOUGH to give one definite answer. For a yes or no question, enough means the answer is always yes or always no. If you can get a yes AND a no, it is not enough. This is where one friendly number does the most damage, because it will hand you a yes every time.
Your challenge for today
Pull up 5 inequality or absolute value questions you already got RIGHT. Redo each one on paper, and for each one test at least three numbers from the edge kit, including the boundary. Then write one line per question: which edge would have broken the trap answer? If you cannot find it on even one of the five, I'll bet you got that question right by luck the first time. Find out today, not on test day.
