Answer:
- excluded values: -2, 2
- LCD: (x -2)(x +2)
- equation: (x+2) +(x -2) = 4
- solution: none
Step-by-step explanation:
A) Excluded Values
Excluded values are values of x that are not in the domain of the given equation. That is, they are values that make the equation "undefined." In general, those are values that make any of the denominators be zero.
x -2 = 0 ⇒ x = 2 must be excluded
x +2 = 0 ⇒ x = -2 must be excluded
x² -4 = (x +2)(x -2) ⇒ x = -2, 2 must be excluded
The excluded values are -2, 2.
__
B) LCD
The least common denominator is the product of the unique denominator factors:
LCD = (x -2)(x +2)
__
C) Equation after Multiplying by the LCD
Multiplying by the LCD gives ...
[tex]\dfrac{1\cdot(x -2)(x+2)}{x-2}+\dfrac{1\cdot(x-2)(x+2)}{x+2}=\dfrac{4\cdot(x-2)(x+2)}{(x-2)(x+2)}\\\\(x+2)+(x-2)=4[/tex]
__
D) Solution
Simplifying the above equation and solving for x gives ...
[tex]2x=4\qquad\text{simplify}\\\\x=2\qquad\text{divide by 2; not an allowed solution}[/tex]
The value we obtain for x is not an allowed value. This equation has ...
no solutions
__
Alternate Solution
If we subtract the right side of the equation from both sides, we get ...
[tex]\dfrac{1}{x-2}+\dfrac{1}{x+2}-\dfrac{4}{(x-2)(x+2)}=0\\\\\dfrac{(x+2)+(x-2)-4}{(x-2)(x+2)}=0\\\\\dfrac{2(x-2)}{(x+2)(x-2)}=0\qquad\text{simplify, factor}\\\\\dfrac{2}{x+2}=0\qquad\text{cancel common factors; this has }\boxed{\textbf{no solution}}[/tex]
There is no value of x that will make 2 = 0 be a true statement.
Note that we have arrived at the conclusion the equation has NO SOLUTION without having to deal with any extraneous solutions.
__
You can see this working in a different context here:
https://brainly.com/question/26248187
