Question

solve \\(\frac{x - 5}{x + 6} = 0\\) and \\(\frac{x - 5}{x + 6} \leq 0\\) analytically. then use a graphing calculator to graph and solve \\(\frac{x - 5}{x + 6} \geq 0\\). select the correct choice below and, if necessary, fill in the answer box to complete your choice. \\(\bigcirc\\) a. the solution set to \\(\frac{x - 5}{x + 6} = 0\\) is \\(\\{\square\\}\\). (use a comma to separate answers as needed.) \\(\bigcirc\\) b. there is no solution.

Explanation:

Step1: Analyze the rational equation

To solve \(\frac{x - 5}{x + 6}=0\), we know that a fraction equals zero when its numerator is zero (and the denominator is not zero, to avoid division by zero).
So we set the numerator equal to zero: \(x - 5 = 0\).

Step2: Solve for \(x\)

Solving \(x - 5 = 0\) gives \(x = 5\).
We also need to check that the denominator is not zero when \(x = 5\). Substituting \(x = 5\) into the denominator \(x + 6\), we get \(5+6 = 11
eq0\), so \(x = 5\) is a valid solution.

Answer:

A. The solution set to \(\frac{x - 5}{x + 6}=0\) is \(\{5\}\).