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avatar+794 

An expression is well-defined if you can compute its value without any illegal operations.  Examples of expressions that are not well-defined include 1/0 and \sqrt{-10}.  For what values of x is the expression
\frac{\sqrt{4x + 1}}{\sqrt{-4x - 1}}
well-defined?

 Dec 29, 2023
 #1
avatar+676 
+1

 

Consider:         sqrt(4x + 1) / sqrt(–4x – 1)    

 

Not Allowed:     denominator equal to zero  

                        negative number under a radical  

 

Denominator:  

x has to be smaller than –1/4   

that is, larger than |1/4| in the negative direction    

to keep the number under the radical greater than zero      

 

Numerator:  

However, any x smaller than –1/4 creates a negative number under the radical    

 

Therefore, there is no solution to this problem;  

that is, there is no value of x that will make the expression well-defined.   

.

 Dec 30, 2023
 #2
avatar+126665 
+1

sqrt [ 4x + 1 ]  / sqrt [ -4x - 1]

 

4x + 1 ≥ 0               -4x - 1 >  0

4x ≥ - 1                   -4x > 1            (divide by -4  and  change the  direction of the inequality sign )

x ≥ -1/4                      x <  - 1/4

 

There are no intersecting intervals....thus.....no values of  x that  make this well-defined

 

cool cool cool

 Dec 30, 2023

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