Solve the inequality problem that involves modulus

chwala
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Homework Statement
See attached
Relevant Equations
modulus
This is the question * consider the highlighted question only *with its solution shown (from textbook);

1639307124007.png


My approach is as follows (alternative method),
##|\frac {5}{2x-3}| ##< ## 1##
Let, ##\frac {5}{2x-3} ##⋅##\frac {5}{2x-3}##=##1##
→##x^2-3x-4=0##
##(x+1)(x-4)=0##
it follows that the critical values are ##x_1=-1## and ##x_2=4##, which will give the correct solution once we confirm/establish the valid region in the given inequality.
Is their another method on this type of questions? ... regards
 
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5&lt;|2x-3|
5/2&lt;|x-3/2|
x-3/2&lt;-5/2, \ 5/2&lt;x-3/2
x&lt;-1, \ 4&lt;x
 
anuttarasammyak said:
5&lt;|2x-3|
5/2&lt;|x-3/2|
x-3/2&lt;-5/2, \ 5/2&lt;x-3/2
x&lt;-1, \ 4&lt;x
Nice one, I will write this in my notebook ...good way to simplify ...##ax...##
 
anuttarasammyak said:
##5<|2x-3|##
##5/2<|x-3/2|##
##x-3/2<-5/2, \ 5/2<x-3/2##
##x<-1, \ 4<x##
The use of commas in the last two lines implies "and", which can't happen here, as it's impossible for x to be both less than -1 and larger than 4.
Here's how I would write this:
##|2x - 3| > 5##
##2x - 3 > 5 \text{ or } 2x - 3 < -5##
##2x > 8 \text{ or } 2x < -2##
##x > 4 \text{ or } x < -1##
 
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Likes chwala and anuttarasammyak
Thanks Mark...a learning point for me too...
Mark44 said:
The use of commas in the last two lines implies "and", which can't happen here, as it's impossible for x to be both less than -1 and larger than 4.
Here's how I would write this:
##|2x - 3| > 5##
##2x - 3 > 5 \text{ or } 2x - 3 < -5##
##2x > 8 \text{ or } 2x < -2##
##x > 4 \text{ or } x < -1##
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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