On December 20, 2023, Brent will stop offering office hours.
- Video Course
- Video Course Overview
- General GMAT Strategies - 7 videos (free)
- Data Sufficiency - 16 videos (free)
- Arithmetic - 38 videos
- Powers and Roots - 36 videos
- Algebra and Equation Solving - 73 videos
- Word Problems - 48 videos
- Geometry - 42 videos
- Integer Properties - 38 videos
- Statistics - 20 videos
- Counting - 27 videos
- Probability - 23 videos
- Analytical Writing Assessment - 5 videos (free)
- Reading Comprehension - 10 videos (free)
- Critical Reasoning - 38 videos
- Sentence Correction - 70 videos
- Integrated Reasoning - 17 videos
- Study Guide
- Blog
- Philosophy
- Office Hours
- Extras
- Prices
Comment on Quadratic and Absolute Value
Hi Brent, first of all thank
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.
Be careful. There are two
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
The number line isn't 100%
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
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
YOUR QUESTION: Why are you
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
There's no general rule; you
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
The rules/strategies won't be
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
If there were MORE THAN ONE
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.
both statements are identical
That's correct.
That's correct.
Hi Brent, for St 1, since I
Be careful; the quadratic
Be careful; the quadratic EQUATION x² - 4x + 3 = 0 is different from the quadratic INEQUALITY x² - 4x + 3 < 0.
In the EQUATION, we are saying that x² - 4x + 3 EQUALS 0.
In the INEQUALITY, we are saying that x² - 4x + 3 does NOT equal 0. Instead, we are saying that x² - 4x + 3 < 0.
Thanks Brent for the
That's correct.
That's correct.