\(\frac{(x^2+6x-7)}{|x+4|} < 0\)

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


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Answer Should be C

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
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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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I think this question can be solved easily with diagram

HTH

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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