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Math: Absolute value (Modulus)

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Re: Math: Absolute value (Modulus) [#permalink]

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New post 21 Jan 2017, 06:55
Bunuel wrote:
gupta87 wrote:
If x is an integer, what is the value of x?
(1) –2(x + 5) < –1
(2) –3x > 9


from S1 it should be x>4.5...........but in explanation its given x>-4.5.
pls explain the logic


–2(x + 5) < –1

Divide by -2 and flip the sign: x + 5 > 0.5;

Deduct 5: x > 0.5 - 5

x > -4.5

Discussed here: if-x-is-an-integer-what-is-the-value-of-x-109983.html

Hope it helps.


understood.but please correct me where am i going wrong??
-2x-10<-1 => -2x<9 =>2x>9 => x>4.5

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Re: Math: Absolute value (Modulus) [#permalink]

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New post 22 Jan 2017, 02:41
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gupta87 wrote:
Bunuel wrote:
gupta87 wrote:
If x is an integer, what is the value of x?
(1) –2(x + 5) < –1
(2) –3x > 9


from S1 it should be x>4.5...........but in explanation its given x>-4.5.
pls explain the logic


–2(x + 5) < –1

Divide by -2 and flip the sign: x + 5 > 0.5;

Deduct 5: x > 0.5 - 5

x > -4.5

Discussed here: if-x-is-an-integer-what-is-the-value-of-x-109983.html

Hope it helps.


understood.but please correct me where am i going wrong??
-2x-10<-1 => -2x<9 =>2x>9 => x>4.5


-2x < 9

Divide by -2 and flip the sign x > -4.5,
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Re: Math: Absolute value (Modulus) [#permalink]

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New post 06 May 2017, 16:54
Thanks for the post. For the 3 step approach for complex problems, how do you find x (E.g. x=-1, x=-15)?

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Re: Math: Absolute value (Modulus) [#permalink]

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New post 11 Oct 2017, 18:08
walker wrote:
ABSOLUTE VALUE
(Modulus)

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This post is a part of [GMAT MATH BOOK]

created by: walker
edited by: bb, Bunuel

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Definition

The absolute value (or modulus) \(|x|\) of a real number x is x's numerical value without regard to its sign.

For example, \(|3| = 3\); \(|-12| = 12\); \(|-1.3|=1.3\)



Graph:
Image


Important properties:

\(|x|\geq0\)

\(|0|=0\)

\(|-x|=|x|\)

\(|x|+|y|\geq|x+y|\)

\(|x|\geq0\)

How to approach equations with moduli

It's not easy to manipulate with moduli in equations. There are two basic approaches that will help you out. Both of them are based on two ways of representing modulus as an algebraic expression.

1) \(|x| = \sqrt{x^2}\). This approach might be helpful if an equation has × and /.

2) |x| equals x if x>=0 or -x if x<0. It looks a bit complicated but it's very powerful in dealing with moduli and the most popular approach too (see below).

3-steps approach:

General approach to solving equalities and inequalities with absolute value:

1. Open modulus and set conditions.
To solve/open a modulus, you need to consider 2 situations to find all roots:
  • Positive (or rather non-negative)
  • Negative

For example, \(|x-1|=4\)
a) Positive: if \((x-1)\geq0\), we can rewrite the equation as: \(x-1=4\)
b) Negative: if \((x-1)<0\), we can rewrite the equation as: \(-(x-1)=4\)
We can also think about conditions like graphics. \(x=1\) is a key point in which the expression under modulus equals zero. All points right are the first condition \((x>1)\) and all points left are second condition \((x<1)\).
Image

2. Solve new equations:
a) \(x-1=4\) --> x=5
b) \(-x+1=4\) --> x=-3

3. Check conditions for each solution:
a) \(x=5\) has to satisfy initial condition \(x-1>=0\). \(5-1=4>0\). It satisfies. Otherwise, we would have to reject x=5.
b) \(x=-3\) has to satisfy initial condition \(x-1<0\). \(-3-1=-4<0\). It satisfies. Otherwise, we would have to reject x=-3.


3-steps approach for complex problems

Let’s consider following examples,

Example #1
Q.: \(|x+3| - |4-x| = |8+x|\). How many solutions does the equation have?
Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:

a) \(x < -8\). \(-(x+3) - (4-x) = -(8+x)\) --> \(x = -1\). We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) \(-8 \leq x < -3\). \(-(x+3) - (4-x) = (8+x)\) --> \(x = -15\). We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) \(-3 \leq x < 4\). \((x+3) - (4-x) = (8+x)\) --> \(x = 9\). We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) \(x \geq 4\). \((x+3) + (4-x) = (8+x)\) --> \(x = -1\). We reject the solution because our condition is not satisfied (-1 is not more than 4)

(Optional) The following illustration may help you understand how to open modulus at different conditions.
Image
Answer: 0

Example #2
Q.: \(|x^2-4| = 1\). What is x?
Solution: There are 2 conditions:

a) \((x^2-4)\geq0\) --> \(x \leq -2\) or \(x\geq2\). \(x^2-4=1\) --> \(x^2 = 5\). x e {\(-\sqrt{5}\), \(\sqrt{5}\)} and both solutions satisfy the condition.

b) \((x^2-4)<0\) --> \(-2 < x < 2\). \(-(x^2-4) = 1\) --> \(x^2 = 3\). x e {\(-\sqrt{3}\), \(\sqrt{3}\)} and both solutions satisfy the condition.

(Optional) The following illustration may help you understand how to open modulus at different conditions.
Image
Answer: \(-\sqrt{5}\), \(-\sqrt{3}\), \(\sqrt{3}\), \(\sqrt{5}\)


Tip & Tricks

The 3-steps method works in almost all cases. At the same time, often there are shortcuts and tricks that allow you to solve absolute value problems in 10-20 sec.

I. Thinking of inequality with modulus as a segment at the number line.

For example,
Problem: 1<x<9. What inequality represents this condition?
Image
A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5
Solution: 10sec. Traditional 3-steps method is too time-consume technique. First of all we find length (9-1)=8 and center (1+8/2=5) of the segment represented by 1<x<9. Now, let’s look at our options. Only B and D has 8/2=4 on the right side and D had left site 0 at x=5. Therefore, answer is D.

II. Converting inequalities with modulus into a range expression.
In many cases, especially in DS problems, it helps avoid silly mistakes.

For example,
|x|<5 is equal to x e (-5,5).
|x+3|>3 is equal to x e (-inf,-6)&(0,+inf)

III. Thinking about absolute values as the distance between points at the number line.

For example,
Problem: A<X<Y<B. Is |A-X| <|X-B|?
1) |Y-A|<|B-Y|
Solution:
Image
We can think of absolute values here as the distance between points. Statement 1 means than the distance between Y and A is less than that between Y and B. Because X is between A and Y, |X-A| < |Y-A| and at the same time the distance between X and B will be larger than that between Y and B (|B-Y|<|B-X|). Therefore, statement 1 is sufficient.


Pitfalls

The most typical pitfall is ignoring the third step in opening modulus - always check whether your solution satisfies conditions.


Official GMAC Books:

The Official Guide, 12th Edition: PS #22; PS #50; PS #130; DS #1; DS #153;
The Official Guide, Quantitative 2th Edition: PS #152; PS #156; DS #96; DS #120;
The Official Guide, 11th Edition: DT #9; PS #20; PS #130; DS #3; DS #105; DS #128;

Generated from [GMAT ToolKit]


Resources

Absolute value DS problems: [search]
Absolute value PS problems: [search]

Fig's post with absolute value problems: [Absolute Value Problems]

A lot of questions as well as a separate topic of PrepGame on absolute value is included in GMAT ToolKit 2

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[Reveal] Spoiler: Images
Attachment:
lineAXYZ.png
Attachment:
line1x9.png
Attachment:
graph_modulus.png
Attachment:
Math_icon_absolute_value.png
Attachment:
Math_abs_example1.png
Attachment:
Math_abs_example2.png
Attachment:
Math_abs_example0.png




II. Converting inequalities with modulus into a range expression.
In many cases, especially in DS problems, it helps avoid silly mistakes.

For example,
|x|<5 is equal to x e (-5,5).
|x+3|>3 is equal to x e (-inf,-6)&(0,+inf)

How did u get -inf and 0 could u pls explain?

Also, how did u get B and D as the last two options?

Only B and D has 8/2=4 on the right side and D had left site 0 at x=5. Therefore, answer is D.

Pls help..

Thanks in advance

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Re: Math: Absolute value (Modulus) [#permalink]

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New post 02 Nov 2017, 02:30
Hi,
In example #1, I am sorry I don't understand the logic to put either + or - after each group of the equation. How do you choose to put + or - before -/+(x+3)-/+(4−x)=+/-(8+x) for each scenario ?

Thanks

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Math: Absolute value (Modulus) [#permalink]

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New post 26 Nov 2017, 06:23
I am Able to solve most of the absolute value problems using the 3 steps approach. But every time I am taking more than 2 minutes to solve them. Is it possible to apply number line method to solve all the modulus problems? Can anyone please explain with some examples how to apply number line method.

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Math: Absolute value (Modulus)   [#permalink] 26 Nov 2017, 06:23

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