Question: Quadratic and Absolute Value

Comment on Quadratic and Absolute Value

Hi Brent, first of all thank you for all the lessons, I am having so much fun preparing for GMAT now.

In this question, in the last step of the second statement you got:

3 <2x <5

After this step you divide all the sides by 2. From the earlier lessons, I have gathered that when we divide inequality the signs change in the reverse direction.

So shouldn't it be 1.5 > x > 2.5 as the final answer. The statement still will be sufficient. But just wanted a clarity on the rules.

Thank you Brent.
gmat-admin's picture

Be careful. There are two rules pertaining to this.

If we divide or multiply both/all sides of an inequality by a POSITIVE value, then the direction of the inequality signs stay the SAME.

If we divide or multiply both/all sides of an inequality by a NEGATIVE value, then the direction of the inequality signs are REVERSED.

More here: https://www.gmatprepnow.com/module/gmat-algebra-and-equation-solving/vid...

Are there any other methods solve the first statement other than checking on the number line?
gmat-admin's picture

The number line isn't 100% necessary for solving these questions. Once you've factored the quadratic and determined the critical points (the values of x such that the quadratic evaluates to be 0), you can just test various values, without plotting the outputs on the number line.

I do not understand statement 2. |2x-4|<1

So shouldnt that be

1) 2x - 4 < 1 -------- x < 2.5
2) 2x - 4 < -1 -------- 2x < 3 ------x < 1.5

Why do you have x > 1.5? Because the negative sign should only go to the right hand side when you open the | | of the left hand side. Why are you multiplying the left hand side by negative also? thanks
gmat-admin's picture

YOUR QUESTION: Why are you multiplying the left hand side by negative also?
I'm not multiplying anything. I'm applying a rule regarding absolute value inequalities, which goes like this:

When solving inequalities involving ABSOLUTE VALUE, there are 2 things you need to know:

- Rule #1: If |something| < k, then –k < something < k
- Rule #2: If |something| > k, then EITHER something > k OR something < -k
(Note: these rules assume that k is positive)

This is covered in the following video: https://www.gmatprepnow.com/module/gmat-algebra-and-equation-solving/vid...

GIVEN: |2x - 4| < 1
From Rule #1, we get: -1 < 2x - 4 < 1
----------------------------

Please note that we can test your theory by plugging in numbers.

For example, if |2x - 4| < 1, then one possible solution is x = 2
When we plug in x = 2, we get: |2(2) - 4| < 1
This simplifies to be |0| < 1, which is true.

You wrote: 2) 2x-4 < -1 ----- 2x < 3 ---- x < 1.5
This tells us that x must be less than 1.5, however we just showed that x = 2 is a possible solution.

Does that help?

Cheers,
Brent

Thanks. Is there a similar rule for squares in inequalities? I mean is that the same rule if it was |2x-4|^2? Or is it different when we deal with squares? thanks
gmat-admin's picture

There's no general rule; you'll have to approach that format on a case-by-case basis.
That said, it would be VERY RARE to encounter a GMAT question involving the square of an absolute value.

Cheers,
Brent

Actually my question is different. I will rephrase. If there we are dealing with squares with inequalities, is there a rule that we can use just like you shared the rule for absolute values? thanks
gmat-admin's picture

The rules/strategies won't be the same.

Consider these examples:
If x² < 9, then we can write: -3 < x < 3
If x² < 49, then we can write: -7 < x < 7
If x² < 100, then we can write: -10 < x < 10

IN GENERAL: If x² < k (where k ≥ 0), then we can write: -√k < x < √k
-------------------------------------

What about when the square is greater than some value?
Consider these examples:
If x² > 9, then we can write: x > 3 or x < -3
If x² > 49, then we can write: x > 7 or x < -7
If x² > 121, then we can write: x > 11 or x < -11

IN GENERAL: If x² > k (where k ≥ 0), then we can write: x > √k or x < -√k

Does that help?

Cheers,
Brent

Thanks so much!

Hi Brent, what if there were more than one value in the middle region? Would the 1st statement be sufficient in that case ?
gmat-admin's picture

If there were MORE THAN ONE integer value within the given range, then statement 1 would be insufficient, since there would be more than one possible value for x.

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