Question
Using the rational expression below, identify the restriction(s) to the domain.
1/x+2+5x/x-4=1/x
-2, 4, 0
2, -4
2, -4, 0
-2, 4
1/x+2+5x/x-4=1/x
-2, 4, 0
2, -4
2, -4, 0
-2, 4
Answers
GPT-4o mini
To identify the restrictions to the domain of the rational expression:
\[
\frac{1}{x+2} + \frac{5x}{x-4} = \frac{1}{x}
\]
we need to determine where the denominators are equal to zero, since division by zero is undefined.
1. The first term, \(\frac{1}{x+2}\), has a denominator of \(x + 2\). This expression is undefined when \(x + 2 = 0\) (i.e., \(x = -2\)).
2. The second term, \(\frac{5x}{x-4}\), has a denominator of \(x - 4\). This expression is undefined when \(x - 4 = 0\) (i.e., \(x = 4\)).
3. The right-hand side, \(\frac{1}{x}\), has a denominator of \(x\). This expression is undefined when \(x = 0\).
Thus, the restrictions on the domain are \(x = -2\), \(x = 0\), and \(x = 4\).
Putting this all together, the restrictions to the domain are:
**-2, 0, 4**
Given the options you provided, it appears there may be a mistake in the provided answers or their formats. However, the correct restrictions are \(-2, 0, 4\).
\[
\frac{1}{x+2} + \frac{5x}{x-4} = \frac{1}{x}
\]
we need to determine where the denominators are equal to zero, since division by zero is undefined.
1. The first term, \(\frac{1}{x+2}\), has a denominator of \(x + 2\). This expression is undefined when \(x + 2 = 0\) (i.e., \(x = -2\)).
2. The second term, \(\frac{5x}{x-4}\), has a denominator of \(x - 4\). This expression is undefined when \(x - 4 = 0\) (i.e., \(x = 4\)).
3. The right-hand side, \(\frac{1}{x}\), has a denominator of \(x\). This expression is undefined when \(x = 0\).
Thus, the restrictions on the domain are \(x = -2\), \(x = 0\), and \(x = 4\).
Putting this all together, the restrictions to the domain are:
**-2, 0, 4**
Given the options you provided, it appears there may be a mistake in the provided answers or their formats. However, the correct restrictions are \(-2, 0, 4\).