A 2 min read

Here’s a GMAT Quant question that transformed our teaching approach and might just change how you tackle absolute value questions forever:

If x is a positive number and |-4x + 8| ≥ 2, which of the following statements represents the exact range of x?

A. 3/2 ≤ x ≤ 5/2

B. 0 < x < 3/2 or x > 5/2

C. x ≤ 3/2 or x ≥ 5/2

D. 0 < x ≤ 3/2 or x ≥ 5/2

E. No solution

• The “Aha!” Moment

A student confidently declared: “Ma’am, why can’t we just split it at x = 2? When x is less than 2, the expression will be negative, and when x is greater than 2, it will be positive.”

I almost agreed… until something stopped me dead in my tracks.

Let’s test this logic with x = 1:

• When x = 1:
• -4(1) + 8 = 4

Wait… 4 is POSITIVE? But x = 1 is less than 2… shouldn’t this be negative?

• The Mind-Bending Revelation

Here’s what makes this problem fascinating: the negative coefficient (-4) completely flips our intuition upside down. Here’s why:

• As x increases, -4x becomes MORE negative
• Therefore, -4x + 8 actually DECREASES as x increases
• Traditional intuition = shattered
  
• Why Students Fall Into The Trap

Nearly 80% of my students pick Choice E (No solution). Here’s their typical thought process:

1. “For x < 2: Expression must be negative” (Wrong!)
2. “For x > 2: Expression must be positive” (Wrong again!)
3. When they solve with these assumptions, they get contradictory results
4. Conclusion: “No solution exists!”

• The GMAT’s Brilliant Twist

Here’s what makes this question genius:

With a positive coefficient (like |2x – 4|):

• As x increases → expression increases
• Split point cleanly divides negative/positive regions

But with a negative coefficient (like |-4x + 8|):

• As x increases → expression DECREASES
• Everything you thought you knew gets reversed!
• The Correct Solution Path

1. Find the zero point: -4x + 8 = 0 → x = 2
2. Critical realization: For x < 2, the expression is POSITIVE (not negative!)
3. For x > 2, the expression is NEGATIVE (not positive!)
4. Solve |-4x + 8| ≥ 2 algebraically

This leads us to: 0 < x ≤ 3/2 or x ≥ 5/2 (Choice D)

• TL;DR

When you see a negative coefficient in an absolute value inequality:

• Your “less than = negative, greater than = positive” instinct will betray you
• The expression DECREASES as x increases
• Don’t fall for the “No solution” trap – that’s exactly what the GMAT wants!

(Pro tip: Don’t let your instincts about positive/negative regions mislead you!)

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Ready to apply this insight? Try this question: If x is positive and |-3x + 12| > 3, find the range of x.