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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
+676
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Consider:         sqrt(4x + 1) / sqrt(–4x – 1)

Not Allowed:     denominator equal to zero

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
+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

Dec 30, 2023