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I learnt this trick while I was in school and yesterday while solving one question I recalled. Its good if you guys use it 1-2 times to get used to it.

Suppose you have the inequality

f(x) = (x-a)(x-b)(x-c)(x-d) < 0

Just arrange them in order as shown in the picture and draw curve starting from + from right.

now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.

Don't forget to arrange then in ascending order from left to right. a<b<c<d

So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d) and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)

If f(x) has three factors then the graph will have - + - + If f(x) has four factors then the graph will have + - + - +

If you can not figure out how and why, just remember it. Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.

For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.

Hi Can you please explain this question to me using the graph. I am missing the point when graph is being used here?

If x is an integer, is |x|>1? (1) (1-2x)(1+x)<0 (2) (1-x)(1+2x)<0

For me ,the first equations roots are -1 and 1/2. Now I am struggling to get to the correct sign using the graph method here.

Same for second equation: roots are 1 and -1/2 but struggling for the sign.

sol +++++ -1 ------ 1/2 +++++ As Karishma pointed out ponce in the req form, the rightmost will always be positive and the alternating will happen from there.

So sol for this x> 1/2 and x<-1

Integers greater than 1/2 and less than -1 , thus |x| may be >= 1. Unsure

Thus Insufficient.

For Statement II

(2) (1-x)(1+2x)<0

Once you get this into the req form

2(x+1/2) (x-1) > 0

sol +++++ -1/2 ------ 1 +++++

So sol for this x>1 and x< -1/2

Integers greater than 1 and less than -1/2 , thus |x| may be >= 1. Unsure

Thus Insufficient.

Combining Both the statements

x<-1 and x>1

Thus Integers for this range will give |x| > 1

Thus Sufficient.

Hope this helps. I am not very confident of my solution though. Its my first solution gmatclub

mayankpant wrote:

gurpreetsingh wrote:

I learnt this trick while I was in school and yesterday while solving one question I recalled. Its good if you guys use it 1-2 times to get used to it.

Suppose you have the inequality

f(x) = (x-a)(x-b)(x-c)(x-d) < 0

Just arrange them in order as shown in the picture and draw curve starting from + from right.

now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.

Don't forget to arrange then in ascending order from left to right. a<b<c<d

So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d) and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)

If f(x) has three factors then the graph will have - + - + If f(x) has four factors then the graph will have + - + - +

If you can not figure out how and why, just remember it. Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.

For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.

Hi Can you please explain this question to me using the graph. I am missing the point when graph is being used here?

If x is an integer, is |x|>1? (1) (1-2x)(1+x)<0 (2) (1-x)(1+2x)<0

For me ,the first equations roots are -1 and 1/2. Now I am struggling to get to the correct sign using the graph method here.

Same for second equation: roots are 1 and -1/2 but struggling for the sign.

VeritasPrepKarishma - Could you please help me find solution for this equation using the graph method?

\(|x^2 - 4| > 3x\). I know Bunuel and others have replied with different solutions but could you show me how this can be solved using graphs?

Draw the diagram. x^2 is a quadratic with minimum at (0, 0). To get x^2 - 4, move the curve down 4 units on the y axis. Take the mod by flipping whatever is below x axis to its reflection above x axis.

Attachment:

Ques3.jpg [ 15.92 KiB | Viewed 6481 times ]

Before the first intersection of the line 3x with the hump of |x^2 - 4|, |x^2 - 4| is greater. After this intersection, the line 3x is greater. Note that 3x will intersect the curve again and after that, |x^2 - 4| will again be greater than 3x. So all we need to do is find the intersections. To get first point of intersection: -(x^2 - 4) = 3x x^2 + 3x -4 = 0 x = 1, -4

To get second point of intersection: x^2 - 4 = 3x x^2 - 3x - 4 = 0 x = 4, -1

Ignore negative x values since our intersections are in first quadrant only. Between x = 1 and x = 4, 3x is greater than |x^2 - 4|. But when x < 1 or x > 4, |x^2 - 4| > 3x.
_________________

I am so very sorry, what I meant to ask was to show me how to find a solution using "plot the roots on the number line" method. I followed your blog approach and somehow I did not end up with the correct answer. I am attaching a screen grab of what I did. Please correct me.

I am so very sorry, what I meant to ask was to show me how to find a solution using "plot the roots on the number line" method. I followed your blog approach and somehow I did not end up with the correct answer. I am attaching a screen grab of what I did. Please correct me.

sol +++++ -1 ------ 1/2 +++++ As Karishma pointed out ponce in the req form, the rightmost will always be positive and the alternating will happen from there.

So sol for this x> 1/2 and x<-1

Integers greater than 1/2 and less than -1 , thus |x| may be >= 1. Unsure

Thus Insufficient.

For Statement II

(2) (1-x)(1+2x)<0

Once you get this into the req form

2(x+1/2) (x-1) > 0

sol +++++ -1/2 ------ 1 +++++

So sol for this x>1 and x< -1/2

Integers greater than 1 and less than -1/2 , thus |x| may be >= 1. Unsure

Thus Insufficient.

Combining Both the statements

x<-1 and x>1

Thus Integers for this range will give |x| > 1

Thus Sufficient.

Hope this helps. I am not very confident of my solution though. Its my first solution gmatclub

Responding to a pm:

Yes, on the whole, the solution is fine.

A couple of things:

Question: Is |x| > 1? Rewrite: Is x>1 or x<-1? x is an integer.

"So sol for this x> 1/2 and x<-1 " should be "So sol for this x> 1/2 OR x<-1 " x cannot be both greater than 1/2 AND less than -1. It will be only one of those two.

x could be 1 here so you are right that this statement alone is not sufficient.

Similar analysis for statement 2 too.

Both together, we get that x > 1 or x < -1 so sufficient together.
_________________

sol +++++ -1 ------ 1/2 +++++ As Karishma pointed out ponce in the req form, the rightmost will always be positive and the alternating will happen from there.

So sol for this x> 1/2 and x<-1

Integers greater than 1/2 and less than -1 , thus |x| may be >= 1. Unsure

Thus Insufficient.

For Statement II

(2) (1-x)(1+2x)<0

Once you get this into the req form

2(x+1/2) (x-1) > 0

sol +++++ -1/2 ------ 1 +++++

So sol for this x>1 and x< -1/2

Integers greater than 1 and less than -1/2 , thus |x| may be >= 1. Unsure

Thus Insufficient.

Combining Both the statements

x<-1 and x>1

Thus Integers for this range will give |x| > 1

Thus Sufficient.

Hope this helps. I am not very confident of my solution though. Its my first solution gmatclub

mayankpant wrote:

gurpreetsingh wrote:

I learnt this trick while I was in school and yesterday while solving one question I recalled. Its good if you guys use it 1-2 times to get used to it.

Suppose you have the inequality

f(x) = (x-a)(x-b)(x-c)(x-d) < 0

Just arrange them in order as shown in the picture and draw curve starting from + from right.

now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution. I m sure I have recalled it fully but if you guys find any issue on that do let me know, this is very helpful.

Don't forget to arrange then in ascending order from left to right. a<b<c<d

So for f(x) < 0 consider "-" curves and the ans is : (a < x < b) , (c < x < d) and for f(x) > 0 consider "+" curves and the ans is : (x < a), (b < x < c) , (d < x)

If f(x) has three factors then the graph will have - + - + If f(x) has four factors then the graph will have + - + - +

If you can not figure out how and why, just remember it. Try to analyze that the function will have number of roots = number of factors and every time the graph will touch the x axis.

For the highest factor d if x>d then the whole f(x) > 0 and after every interval of the roots the signs will change alternatively.

Hi Can you please explain this question to me using the graph. I am missing the point when graph is being used here?

If x is an integer, is |x|>1? (1) (1-2x)(1+x)<0 (2) (1-x)(1+2x)<0

For me ,the first equations roots are -1 and 1/2. Now I am struggling to get to the correct sign using the graph method here.

Same for second equation: roots are 1 and -1/2 but struggling for the sign.

THanks

Thanks this helps. I was not bringing the equation tot he correct form.

I had read this post a while ago, and not heeded the advice (i thought it was beyond my understanding). But I came across quite a few questions where this could be used (links that I saw in the discreet charm of DS post), and realized knowing this could be blessing. But to make sure i understood right, here's a paraphrase: In cases of inequality regarding factors of a function, in terms of zero and depending on the sign of inequality, a few shortcuts (of which, the logic has been explained by veritasprepkarishma) can be employed. The roots of the inequality, once arranged in the ascending order, maybe on a number line even, can be assigned + and - signs by starting from + in the right most portion of the line(which is open-ended), alternating between +and - till the last portion (i.e. the left most one) Now, here is where the sign '<'/'>' matters: if the sign is < then I am to consider the ranges where the - sign lies (set by the roots marking such a range) and if the sign is > then i am to consider ranges that contain + sign. -Is this correct? -This applies only for inequalities of zero, right? -Also, in the "now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution.....", what should I combine? I don't think we are talking about combining + and - here, are we? Am I to combine the ranges and place the variable(in this case 'x') in this range? Thanks in advance

I had read this post a while ago, and not heeded the advice (i thought it was beyond my understanding). But I came across quite a few questions where this could be used (links that I saw in the discreet charm of DS post), and realized knowing this could be blessing. But to make sure i understood right, here's a paraphrase: In cases of inequality regarding factors of a function, in terms of zero and depending on the sign of inequality, a few shortcuts (of which, the logic has been explained by veritasprepkarishma) can be employed. The roots of the inequality, once arranged in the ascending order, maybe on a number line even, can be assigned + and - signs by starting from + in the right most portion of the line(which is open-ended), alternating between +and - till the last portion (i.e. the left most one) Now, here is where the sign '<'/'>' matters: if the sign is < then I am to consider the ranges where the - sign lies (set by the roots marking such a range) and if the sign is > then i am to consider ranges that contain + sign. -Is this correct?

Yes, but remember, the factors of the inequality should be of the form (ax-b), (cx-d) etc. If they are not in this form, convert them to this form using the techniques explained in the links given here: inequalities-trick-91482-60.html#p1265455

Quote:

-This applies only for inequalities of zero, right?

Yes, but you will be able to bring most GMAT inequalities in the 'zero on the right hand side' form.

Quote:

-Also, in the "now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution.....", what should I combine? I don't think we are talking about combining + and - here, are we? Am I to combine the ranges and place the variable(in this case 'x') in this range? Thanks in advance

Are you talking about a particular question here? Kindly give the link.
_________________

VeritasPrepKarishma has given a very lucid explanation of how this “wavy line” method works.

I have noticed that there is still a little scope to take this discussion further. So here are my two cents on it.

I would like to highlight an important special case in the application of the Wavy Line Method

When there are multiple instances of the same root:

Try to solve the following inequality using the Wavy Line Method:

\((x-1)^2(x-2)(x-3)(x-4)^3 < 0\)

To know how you did, compare your wavy line with the correct one below.

Did you notice how this inequality differs from all the examples above?

Notice that two of the four terms had an integral power greater than 1.

How to draw the wavy line for such expressions?

Let me directly show you how the wavy line would look and then later on the rule behind drawing it.

Attachment:

File comment: Observe how the wave bounces back at x = 1.

bounce.png [ 10.4 KiB | Viewed 6503 times ]

Notice that the curve bounced down at the point x = 1. (At every other root, including x = 4 whose power was 3, it was simply passing through them.)

Can you figure out why the wavy line looks like this for this particular inequality?

(Hint: The wavy line for the inequality

\((x-1)^{38}(x-2)^{57}(x-3)^{15}(x-4)^{27} < 0\)

Is also the same as above)

Come on! Give it a try.

If you got it right, you’ll see that there are essentially only two rules while drawing a wavy line. (Remember, we’ll refer the region above the number line as positive region and the region below the number line as negative region.)

How to draw the wavy line?

1. How to start: Start from the top right most portion. Be ready to alternate (or not alternate) the region of the wave based on how many times a point is root to the given expression.

2. How to alternate: In the given expression, if the power of a term is odd, then the wave simply passes through the corresponding point (root) into the other region (to –ve region if the wave is currently in the positive region and to the +ve region if the wave is currently in the negative region). However, if the power of a term is even, then the wave bounces back into the same region.

Now look back at the above expression and analyze your wavy line. Were you (intuitively) using the above mentioned rules while drawing your wavy line?

Solution

Once you get your wavy line right, solving an inequality becomes very easy. For instance, for the above inequality, since we need to look for the space where the above expression would be less than zero, look for the areas in the wavy line where the curve is below the number line.

So the correct solution set would simply be {3 < x < 4} U {{x < 2} – {1}}

In words, it is the Union of two regions – region1between x = 3 and x = 4 and region2 which is x < 2, excluding the point x = 1.

Food for Thought

Now, try to answer the following questions:

1. Why did we exclude the point x = 1 from the solution set of the last example? (Easy Question) 2. Why do the above mentioned rules (especially rule #2) work? What is/are the principle(s) working behind the curtains?

Foot Note: Although the post is meant to deal with inequality expressions containing multiple roots, the above rules to draw the wavy line are generic and are applicable in all cases.

I had read this post a while ago, and not heeded the advice (i thought it was beyond my understanding). But I came across quite a few questions where this could be used (links that I saw in the discreet charm of DS post), and realized knowing this could be blessing. But to make sure i understood right, here's a paraphrase: In cases of inequality regarding factors of a function, in terms of zero and depending on the sign of inequality, a few shortcuts (of which, the logic has been explained by veritasprepkarishma) can be employed. The roots of the inequality, once arranged in the ascending order, maybe on a number line even, can be assigned + and - signs by starting from + in the right most portion of the line(which is open-ended), alternating between +and - till the last portion (i.e. the left most one) Now, here is where the sign '<'/'>' matters: if the sign is < then I am to consider the ranges where the - sign lies (set by the roots marking such a range) and if the sign is > then i am to consider ranges that contain + sign. -Is this correct?

Yes, but remember, the factors of the inequality should be of the form (ax-b), (cx-d) etc. If they are not in this form, convert them to this form using the techniques explained in the links given here: inequalities-trick-91482-60.html#p1265455

Quote:

-This applies only for inequalities of zero, right?

Yes, but you will be able to bring most GMAT inequalities in the 'zero on the right hand side' form.

Quote:

-Also, in the "now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution.....", what should I combine? I don't think we are talking about combining + and - here, are we? Am I to combine the ranges and place the variable(in this case 'x') in this range? Thanks in advance

Are you talking about a particular question here? Kindly give the link.

When (ax-b) is NOT the form, convert it to such a form, and the root will be b/a. Got it. And no. I was referring to this post itself. I mean, the first post of this thread.

When (ax-b) is NOT the form, convert it to such a form, and the root will be b/a. Got it. And no. I was referring to this post itself. I mean, the first post of this thread.

Yes, you will take the constant a out and will be left with (x - b/a) as a factor. Also, if a factor is (b - x), multiply both sides by -1 to get (x - b). The inequality sign will flip in this case. These and more complications are discussed in the links mentioned in my post above.
_________________

-Also, in the "now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution.....", what should I combine? I don't think we are talking about combining + and - here, are we? Am I to combine the ranges and place the variable(in this case 'x') in this range? Thanks in advance

Are you talking about a particular question here? Kindly give the link.

And no. I was referring to this post itself. I mean, the first post of this thread.

-Also, in the "now if f(x) < 0 consider curve having "-" inside and if f(x) > 0 consider curve having "+" and combined solution will be the final solution.....", what should I combine? I don't think we are talking about combining + and - here, are we? Am I to combine the ranges and place the variable(in this case 'x') in this range?

Hi Guys, I'm in the process of absorbing the fundamentals and neat tricks provided by the experts here. This inequality trick is phenomenal. Would it be safe to conclude that - for odd number of factors or roots, we start with "-" (cosine waveform) and for even number of roots we start with "+" (sine wave)?

As explained by Gurpreet & Karishma: If f(x) has three factors then the graph will have - + - + If f(x) has four factors then the graph will have + - + - +

Similarly: If f(x) has two factors then the graph will have + - + If f(x) has seven factors then the graph will have - + - + - + - + ?

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