To solve the given inequality [tex]\(\frac{x}{x-6} > 13\)[/tex], let's follow these steps:
1. Rewrite the Inequality:
[tex]\[
\frac{x}{x-6} > 13
\][/tex]
2. Eliminate the Denominator:
To eliminate the denominator, multiply both sides by [tex]\((x-6)\)[/tex]. But since [tex]\(x-6\)[/tex] could be positive or negative, we need to consider two cases based on the sign of [tex]\(x-6\)[/tex].
[tex]\[
\text{Case 1: } x > 6 \quad \text{(positive denominator)}
\][/tex]
[tex]\[
\frac{x}{x-6} > 13 \implies x > 13(x - 6)
\][/tex]
[tex]\[
x > 13x - 78
\][/tex]
[tex]\[
x - 13x > -78
\][/tex]
[tex]\[
-12x > -78
\][/tex]
[tex]\[
x < \frac{78}{12} \quad \implies x < 6.5
\][/tex]
Therefore, for [tex]\(x > 6\)[/tex], we have [tex]\(6 < x < 6.5\)[/tex].
[tex]\[
\text{Case 2: } x < 6 \quad \text{(negative denominator)}
\][/tex]
[tex]\[
\frac{x}{x-6} > 13 \implies x < 13(x - 6)
\][/tex]
[tex]\[
x < 13x - 78
\][/tex]
[tex]\[
x - 13x < -78
\][/tex]
[tex]\[
-12x < -78
\][/tex]
[tex]\[
x > \frac{78}{12} \quad \implies x > 6.5
\][/tex]
However, since [tex]\(x\)[/tex] cannot be both greater than 6.5 and less than 6 at the same time, this case does not provide a valid solution. Therefore, there is no solution for [tex]\(x < 6\)[/tex].
3. Combine Results:
After reviewing the cases, the valid interval where [tex]\(\frac{x}{x-6} > 13\)[/tex] holds true is [tex]\(6 < x < 6.5\)[/tex].
Thus, the correct inequality describing the number of seconds Charlene can expect to skate before losing her footing is:
[tex]\[
\boxed{6 < x < 6.5}
\][/tex]
The correct answer is:
C. [tex]\(6 < x < 6.5\)[/tex]
