Absolute fundamentals

Question

Hi

if i have the below equation

| 3 + 2x | > |4 - x |

are there 2 cases o3 3 cases i need to consider to solve for x

3 + 2x > 4 - x

3 + 2x < -4 + x

is there another case i need to consider here ?

KarishmaB · Accepted Answer

Bunuel has provided you the algebraic solution. Let me add the graphical approach. It may seem daunting at first, but if you take the effort of understanding it, it will seem very straight forward and easy. Draw the graphs of both the mods y = | 3 + 2x | and y = |4 - x | as shown below. Ques2.jpg Now where is the graph of | 3 + 2x | > the graph of |4 - x | ? Where is the Red line above the Purple line? Can I say it's so for the values of x as depicted by the green arrows? Now all we need to do is find these points. Point A: 2x+3 = 4-x which implies x = 1/3 Point B: -2x - 3 = 4-x which implies x = -7 So the given condition is satisfied when x > 1/3 or x < -7 Note: If you are wondering how do we decide whether we need to take (2x+3) or (-2x-3), (4-x) or (x-4) while finding points A and B, notice that we need to find the intersection of 2 lines to find these points. When we make the graph of | 3 + 2x |, one line is (3+2x), and the other line is (-3-2x). When we make graph of |4 - x |, one line is (4-x) and the other is (x-4). While finding point A, the Red line is going up from left to right so co-efficient of x must be positive hence we use (2x+3). The purple line is going down from left to right so the co-efficient of x must be negative so we use (4-x). Similarly for point B, the red line is going down from left to right so co-efficient of x must be negative so we use (-2x-3). The purple line is going down from left to right so the co-efficient of x must be negative so we use (4-x).

gmat1220 · Answer

Only two cases have to be considered.

case 1
------
3 + 2x > 4 - x
3x > 1
x > 1/3

case 2
------
3 + 2x > x - 4
x > -7

Taking the most restricted value x > 1/3

Bunuel · Answer

Above solution is not correct. |3+2x|> |4-x| " First you should determine the check points (key points are the values of x for which the expressions in absolute value equal to zero). So the key points are -\frac{3}{2} and 4 . Hence we'll have three ranges to check: A. x<-\frac{3}{2} --> -(3+2x)>{4-x} --> x<{-7} ; B. -\frac{3}{2}\leq{x}\leq{4} --> 3+2x>4-x --> x>\frac{1}{3} , as we consider the range -\frac{3}{2}\leq{x}\leq{4} , then \frac{1}{3}<{x}\leq{4} ; C. x>4 --> 3+2x>-(4-x) --> x>{-7} , as we consider the range x>4 , then x>4 ; Ranges from A, B and C give us the solution as: x<{-7} or x>\frac{1}{3} (combined range from B and C). Similar problem: inequalities-challenging-and-tricky-one-89266.html

gmat1220 · Answer

Bunuel

I got this thoroughly. Please enlighten me on these statements -

a) For a quadratic equation - the value is positive beyond the roots and negative between the roots. In other words the value of the quadratic alternates +/- in between the root intervals.
b) For an inequality - you must examine the root intervals. "sign" of the inequality MAY flip between the root intervals. To determine the exact sign - use the numberline.

I hope I have absorbed the information flawlessly.

thanks

gmat1220 · Answer

+1 This is epic. I wish you taught me inequalities back in school. :-D

KarishmaB · Answer

An even shorter and more intuitive approach to these questions.
Rephrase the question as:
 
| 2x + 3| - |4 - x | > 0
 
2| x + 3/2| - |x - 4 | > 0

The '2' outside the mod means 'twice the distance from -3/2'. Also, |4 - x | is the same as |x - 4 |.

Then see if you can figure out the answer from the number line. (I must tell you that it takes some effort to figure out the first time... A few people in my batches have done it though so its not terribly difficult...) 
I could give an explanation later if you want to verify...

144144 · Answer

Karishma, as usual - u are great. I would be more than happy to learn more about it.

thanks! +1

KarishmaB · Answer

Check out the posts on the blog given in my signature below.
They will help you get comfortable with the basics... Get back in case you have any doubts... Then try and solve this question.... I will give the solution using the method mentioned in the post if you like then....

gmat1220 · Answer

Karishma, 
I have read your blog. Pls verify this - I am doing it intuitively.
If a - b > 0 this means at some point a = b. Lets determine that point. a = |2x+3| and b = |4-x|

Solving -
|2x+3| = |4-x|

Case 1 
------
2x + 3 = 4 - x
3x = 1
x = 1/3
Therefore to make a > b, x > 1/3. i.e. move x further right of zero.

Case 2
-------
2x + 3 = x - 4
x + 7 = 0
x = -7

Therefore to make a > b, x < -7. i.e. move x further left of zero.

KarishmaB · Answer

This is absolutely fine but I would not worry about equating... I would let common sense guide me... Let me tell you what I have in mind Let's focus on just the x axis i.e. the number line Since you have gone through the post, you know that mod is nothing but distance from 0 on the number line... |x| = 2 means 'x is at a distance of 2 from point 0 on the number line' |x-4| = 6 means 'the distance of x from 4 on the number line is 6' so x must be 10 or -2 These are the basics. Let's take an easier example first. What does the following mean? |x+2| > |x-4| It means the value of x is such that distance from -2 is greater than distance from 4. Ques3.jpg Where is the distance from -2 equal to distance from 4? At point 1 on the number line, right? Red and green arrows will be equal. So if x > 1, red arrow will be longer than green i.e. the distance of the point from -2 will be more than the distance from 4. So x > 1 satisfies this inequality. What about points on the left of -2? Will there be any point such that its distance from -2 is equal to the distance from 4? Obviously not. All points will be closer to -2 than to 4. Hence the only region is x > 1. Now let's take the question at hand: |2x+3| > |4-x| 2|x+3/2| > |x-4| It means the value for x is such that twice the distance from -3/2 is more than the distance from 4. Where will twice the distance from -1.5 be exactly equal to distance from 4? Ques2.jpg We should divide the distance of 5.5 between them in 3 equal parts to get 5.5/3 Now lets go 5.5/3 ahead of -1.5 to get -1.5+5.5/3 = 1/3. This is the point where twice the distance from -3/2 is equal to distance from 4. So you go to the right to make twice the distance from -3/2 greater than the distance from 4. So one solution is x > 1/3 Is there some other point where the same thing will happen? Yes, at x = -7. How do I get it? because distance between -1.5 and 4 is 5.5 When I add this to the left of -1.5, I get point 7 which is where double the distance from -1.5 will be equal to distance from point 4. To the left of -7, twice the distance from -1.5 will be greater than the distance from 4. So another solution is x < -7 It is much more intuitive and all you need to do is draw a number line and then reason it out. Let me warn you, it's not everyday that I come across people who are interested in and appreciate alternative strategies. So when I do get an audience, I tend to get a little out of hand... If it makes sense to you, go ahead and try it out.. let me know if you get stuck with anything.. if it doesn't make sense, ignore it...

tinki · Answer

Hello, here is my solution: testing all possibilities with signs

1) 3+2x>4-x
x>1/3
2) -3-2x>-4+x
x<1/3
3) -3-2x>4-x
-7>x
4)3+2x>4+x
x>-7

but i have read somewhere that we shall only test both positive and positive neg. 
so we have solution x>1/3 x>-7 is this correct ? 
thx

windofchange · Answer

My solution:
Because both  |3+2x|  and  |4-x| >= 0 , I can square the two sides:
 (3+2x)^2 > (4-x)^2 
 9+12x+4x^2 > 16-8x+x^2 
 3x^2+20x-7 > 0 
 x1=\frac{1}{3} 
 x2={-7} 
=> we have two ranges:  x<{-7}  or  x>\frac{1}{3}

windofchange · Answer

I think this table will help illustrate the solution of Bunuel.
1.  x<-\frac{3}{2} 
 |3+2x|= -(3+2x)  and  |4-x|={4-x}

2.  -\frac{3}{2}\leq{x}\leq{4} 
 |3+2x|={3+2x}  and  |4-x|={4-x}

3.  x>4 
 |3+2x|={3+2x}  and  |4-x|=-(4-x)

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