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iikarthik wrote:Hi Bunuel,
How do we solve inequalities of the form ax^2 + bx + c >= Y
We have seen examples for > or < but not for >= or <=.
Will there be any change in values or the steps involved in calculation?
Regards,
karthik
Feb 17, 2016
May 31, 2016
Bunuel wrote:Inequalities with Complications - Part IBY KARISHMA, VERITAS PREP
Above we learned how to handle inequalities with many factors i.e. inequalities of the form . This week, let’s see what happens in cases where the inequality is not of this form but can be manipulated and converted to this form. We will look at how to handle various complications.
Complication No. 1:
We want our inequality to be of the form , not because according to the logic we discussed last week, when x is greater than a, we want this factor to be positive. The manipulation involved is pretty simple:
So we get:
But how do we handle the negative sign in the beginning of the expression? We want the values of x for which the negative of this expression should be positive. Therefore, we basically want the value of x for which this expression itself (without the negative sign in the beginning) is negative.
We can manipulate the inequality to
Or simply, multiply by -1 on both sides. The inequality sign flips and you get
e.g. Given:
We can re-write this as
(multiplying both sides by -1)
Now the inequality is in the desired form.
Complication No 2: (where m is a positive constant)
How do we bring to the form ? By taking m common!
The constant does not affect the sign of the expression so we don’t have to worry about it.
e.g. Given:
We can re-write this as
When considering the values of x for which the expression is negative, 2 has no role to play since it is just a positive constant.
Now let’s look at a question involving both these complications.
Question 1: Find the range of x for which the given inequality holds.
Solution:
Given:
(taking x common)
(factoring the quadratic)
(take 2 common)
(multiply both sides by -1)
This inequality is in the required form. Let’s draw it on the number line.
We are looking for negative value of the expression. Look at the ranges where we have the negative sign.
The ranges where the expression gives us negative values are and x < 0.
Hence, the inequality is satisfied if x lies in the range or in the range .
Plug in some values lying in these ranges to confirm.
In the next post, we will look at some more variations which can be brought into this form.
Updated on: Jun 5, 2016
Jun 5, 2016
22gmat wrote:Is it possible that the last graph in your third post is wrong? In my opinion the graph should be positive between 2,5 und 6 and negative for x>6 and x<0?
Thanks!
Jun 5, 2016
Bunuel wrote:Inequalities with Complications - Part IIBY KARISHMA, VERITAS PREP
Now let’s look at a question involving both these complications.
Question: Find the range of x for which the given inequality holds.
Solution:
I hope you agree that it doesn’t matter whether the factors are multiplied or divided. We are only concerned with the sign of the factors.
(take 2 common)
(multiply both sides by -1)
Now let’s draw the number line. We don’t need to plot 0 since (x – 0) has an even power.
We want to find the range where the expression is positive. The required range is or . But we are missing something here. implies that all values less than 5/2 are acceptable but note that x cannot be 0 since x^2 is in the denominator. Hence the acceptable range is or but .
When you have the equal to sign, you have to be careful about the way you choose your range.
Jun 15, 2017
Jul 30, 2017
Bunuel wrote:Questions on InequalitiesBY KARISHMA, VERITAS PREP
Now that we have covered some variations that arise in inequalities in GMAT problems, let’s look at some questions to consolidate the learning.
We will first take up a relatively easy OG question and then a relatively tougher question which looks harder than it is because of the use of mods in the options (even though, we don’t really need to deal with the mods at all).
Question 1: Is n between 0 and 1?
Statement 1: n^2 is less than n
Statement 2: n^3 is greater than 0
Solution: Let’s take each statement at a time and see what it implies.
Statement 1:
This is the required form of the expression. We can now put it on the number line.
For the expression to be negative, n should be between 0 and 1. So we can answer the question with a ‘yes’. Statement 1 alone is sufficient.
Statement 2:
This only implies that n > 0 and we do not know whether it is less than 1 or not. Hence this statement alone is not sufficient.
Answer: (A) This question is discussed HERE.
This question could have been easily solved in a minute if you understand the theory we have been discussing for the past few weeks. Let’s go on to the trickier question now.
Question 2: Which of the following represents the complete range of x over which ?
(A) 0 < |x| < ½
(B) |x| > ½
(C) –½ < x < 0 or ½ < x
(D) x < –½ or 0 < x < ½
(E) x < –½ or x > 0
Solution: As I said, it looks harder than it is. We can easily do this in a minute too. First, let’s look at the given inequality closely:
(taking x^5 common)
Just to make things easier right away, take out 4 common and multiply both sides by -1 to get
(notice that the sign has flipped since we multiplied both sides by -1)
Think of the points you are going to plot: 0, 1/2 and -1/2. Recall that any positive odd power can be treated as a power of 1.
In which region is x positive? or .
This is our option (C). This question is discussed HERE.
A quick word on the other options: What does 0 < |x| < ½ imply? It implies that distance of x from 0 is less than ½. So x lies between -1/2 and 1/2 (but x cannot be 0).
What does imply? It implies that distance of x from 0 is more than 1/2. So x is either greater than 1/2 or less than -1/2.
If you are wondering what I am talking about, check out an old QWQW post: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html/2011/01 ... edore-did/
We have discussed how to deal with modulus here. We hope this discussion has made such questions easier for you!
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