To identify the domain restrictions on the given rational equation:
[tex]\[
\frac{5x}{4x + 40} + \frac{x + 100}{6} = \frac{30}{x^2 - 100}
\][/tex]
we need to determine the values of [tex]\( x \)[/tex] that make any denominator in the equation equal to zero. When a denominator is zero, the equation becomes undefined, which results in the restrictions of the domain.
Step 1: Analyze the first denominator: [tex]\( 4x + 40 \)[/tex]
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
4x + 40 = 0
\][/tex]
[tex]\[
4x = -40
\][/tex]
[tex]\[
x = -10
\][/tex]
This tells us that [tex]\( x = -10 \)[/tex] is a restriction (option A).
Step 2: Analyze the denominator of the right-hand side: [tex]\( x^2 - 100 \)[/tex]
Set the denominator equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 - 100 = 0
\][/tex]
[tex]\[
x^2 = 100
\][/tex]
[tex]\[
x = \pm \sqrt{100}
\][/tex]
[tex]\[
x = \pm 10
\][/tex]
This gives us two more restrictions: [tex]\( x = 10 \)[/tex] and [tex]\( x = -10 \)[/tex] (options B and A, considering we obtained [tex]\( x = -10 \)[/tex] previously).
Step 3: Analyze the second term denominator: [tex]\( 6 \)[/tex]
Note that [tex]\( 6 \)[/tex] is a constant and does not affect the restrictions since it cannot be zero.
Conclusion:
The restrictions for this equation are [tex]\( x = -10 \)[/tex] and [tex]\( x = 10 \)[/tex].
So, selecting the appropriate restrictions from the provided options:
A. [tex]\( x = -10 \)[/tex]
B. [tex]\( x = 10 \)[/tex]
The result is that the correct domain restrictions are options A and B.
