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(x^2+6x-7)/|x+4| < 0

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Narenn wrote:

First of all, equation within Modulus can not be negative. So |x+4| has to be positive.

Now if $$\frac{a}{b}$$ is negative and b is positive, then a must be negative. i.e. $$x^2 + 6x - 7 < 0$$

So we have that $$x^2 + 6x - 7 < 0$$ --------> (x+7)(x-1) < 0 -------> Now we get three ranges of x
1) x < -7
2) -7 < x < 1
3) x > 1
Here we can see that the range mentioned in serial number 2 satisfies the inequality (x+7)(x-1) < 0. So -7 < x < 1

Now let's work on denominator. We know that |x+4| > 0 --------> x > -4 or x < -4 that means x is not equal to -4 -------[ |x-a| > r then x > a+r or x < a-r ]

Consider both the ranges together
-7 < x < 1 and x is not equal to -4
So -7 < x < -4 and -4 < x < 1

Should you want to review basics concepts of Inequations and Modules, Visit my this Article. And this too.

Hope that helps

Hi Narenn,

Here we can see that the range mentioned in serial number 2 satisfies the inequality (x+7)(x-1) < 0. So -7 < x < 1

I have a doubt.

Now, (x+7)(x-1) < 0
CASE 1: x+7 is +ve and x-1 is negative.

x+7 >0 and x -1 <0
x > -7 and x< 1

CASE 2: x+7 is -ve and x-1 is +ve

x+7 <0 and x-1 >0
x< -7 and x> 1

Why haven't you consider the second case???

Thanks,
Jai
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jaituteja wrote:
Narenn wrote:

First of all, equation within Modulus can not be negative. So |x+4| has to be positive.

Now if $$\frac{a}{b}$$ is negative and b is positive, then a must be negative. i.e. $$x^2 + 6x - 7 < 0$$

So we have that $$x^2 + 6x - 7 < 0$$ --------> (x+7)(x-1) < 0 -------> Now we get three ranges of x
1) x < -7
2) -7 < x < 1
3) x > 1
Here we can see that the range mentioned in serial number 2 satisfies the inequality (x+7)(x-1) < 0. So -7 < x < 1

Now let's work on denominator. We know that |x+4| > 0 --------> x > -4 or x < -4 that means x is not equal to -4 -------[ |x-a| > r then x > a+r or x < a-r ]

Consider both the ranges together
-7 < x < 1 and x is not equal to -4
So -7 < x < -4 and -4 < x < 1

Should you want to review basics concepts of Inequations and Modules, Visit my this Article. And this too.

Hope that helps

Hi Narenn,

Here we can see that the range mentioned in serial number 2 satisfies the inequality (x+7)(x-1) < 0. So -7 < x < 1

I have a doubt.

Now, (x+7)(x-1) < 0
CASE 1: x+7 is +ve and x-1 is negative.

x+7 >0 and x -1 <0
x > -7 and x< 1

CASE 2: x+7 is -ve and x-1 is +ve

x+7 <0 and x-1 >0
x< -7 and x> 1

Why haven't you consider the second case???

Thanks,
Jai

We are looking for the value(s) of x that would satisfy the the inequality (x+7)(x-1)<0

Now we know that (x+7)(x-1) will be negative when any one of them be negative and other one be positive. For this to happen we should find the value(s) of x that would simultaneously make one positive and other one negative.

We got two critical points (or check points, or reference points, whatever you call to it) as -7 and 1. We should carry out our search for the right value considering these two reference points.

-Infinity,........-13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, .......... Infinity

Consider the right side of the number line. You will notice that every value beyond critical point 1 makes both the equations positive. So the values beyond 1 do not agree with the inequality and hence can not be the solution of it.

Similarly every value beyond -7 towards negative side makes both the equations negative, thereby making the multiplication (x+7)(x-1) positive. So these values also can not be the solution of the inequality.

Whereas every value between -7 and 1 makes one of the equation positive and at the same time other one negative. So these values can satisfy the inequality and hence are the solution.

You will also notice that the reverse is true when the inequation has greater than(>) sign. That means in such case values between the critical points will not satisfy the inequality and those beyond the critical points would.

You can generalize this as for the quadratic inequation of the form ax^2 + bx + c < 0 the solution lies between the roots and for the quadratic inequation of the form ax^2 + bx + c > 0 the solution lies beyond the roots
(Note :- There is one exception for this rule that is in quadratic equation ax^2 + bx + c < or > 0, if discriminant i.e. b^2 - 4ac is equal to zero then the quadratic equation will give only one root (Critical Point). All values of x except for root(critical point) will satisfy the inequality ax^2 + bx + c > 0 and the inequality ax^2 + bx + c < 0 will not have any solution. Check this example.

Further to understand quadratic inequalities in depth try these methods proposed by some legendary members
BUNUEL
Veritas Karishma
Zarrolou

Hope that Helps!
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Thank you all. The replies are helpful..I will review the posts quoted here first before solving more problems.
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I think this question can be solved easily with diagram

HTH
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(x+7)*(x-1)<0

x<-7 AND x>1=Nothing

x>-7 AND x<1=-7<x<1

test denominator for 0: only -4 gives |x+4|=0

so, -7<x<-4 AND -4<x<1
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I have a question. Since we know that |x+4| is always positive, why can't we multiply both sides of the inequality by |x+4|, to arrive at the following:

x^2 + 6x - 7 < 0
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sauravpaul wrote:
I have a question. Since we know that |x+4| is always positive, why can't we multiply both sides of the inequality by |x+4|, to arrive at the following:

x^2 + 6x - 7 < 0

Yes, you can do that and every solution above does this in one sense or another.
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(x2+6x−7)|x+4|<0

$$﻿\frac{\left(x^2+6x-7\right)}{\left|x+4\right|}\ <\ 0$$
﻿The function inside the modulus is always non-negative.
﻿Hence for the fraction to be negative. The case is possible only when the numerator is negative.
﻿The numerator can be expressed as :
$$﻿x^2+6x-7\ =\ \left(x+7\right)\cdot\left(x-1\right)$$
﻿For this to be negative the possible range of x should be :
﻿-7 < x < 1.
﻿But since for x = -4 the denominator becomes 0. The function is not defined when x = -4.
﻿Hence the range of values of x :
﻿-7 < x < -4 and -4 < x < 1.
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Juggernaut23 wrote:
$$\frac{(x^2+6x-7)}{|x+4|} < 0$$

A) -7<x<-5 and -4<x<1
B) -7<x<-5 and -4<x<0
C) -7<x<-4 and -4<x<1
D) -7<x<-6 and -4<x<1

I am looking for a good way to approach inequality problems that involve absolute value ratios, as given above. Any help is greatly appreciated.

SOURCE: this is not from GMAT prep book; this problem is from a GRE practice book, which have many problems of this kind. I went through M-GMAT book inequalities chapter but haven't seen anything similar to this.

Thanks!

Note that we will use the same wavy line method as we do for inequalities with multiple factors, just that x = -4 will not be a transition point because the inequality does not change sign at x = -4. The absolute value ensures that |x + 4| is always positive so in a way it is like our squares or positive constants. The only thing is that since it is in the denominator, we know that x cannot be -4.

(x^2+6x-7) = (x + 7)(x - 1)
So transition points are -7 and 1 and since we are looking for negative values of inequality, the range we get is -7 < x < 1. But -4 is a part of this range hence we need to remove it.
-7<x<-4 and -4<x<1 is one way of doing it.

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