Question

solve the rational equation. express numbers as integers or simplified fracti
\\(\frac{3}{w^2 + 5w} + \frac{w}{w + 5} = \frac{1}{w}\\)
the solution set is \\(\\{\square\\}\\).

Explanation:

Step1: Factor denominator

First, factor \(w^2 + 5w\) into \(w(w+5)\). The equation becomes:
$$\frac{3}{w(w+5)} + \frac{w}{w+5} = \frac{1}{w}$$

Step2: Eliminate denominators

Multiply all terms by \(w(w+5)\) (the LCD) to cancel denominators:
$$3 + w \cdot w = 1 \cdot (w+5)$$

Step3: Simplify equation

Expand and rearrange into standard quadratic form:
$$3 + w^2 = w + 5$$
$$w^2 - w - 2 = 0$$

Step4: Factor quadratic

Factor the quadratic expression:
$$(w-2)(w+1) = 0$$

Step5: Solve for w

Set each factor equal to 0 and solve:
$$w-2=0 \implies w=2$$
$$w+1=0 \implies w=-1$$

Step6: Check for extraneous solutions

Verify \(w=2\) and \(w=-1\) do not make original denominators 0. Both are valid (denominators are non-zero for these values).

Answer:

The solution set is \(\{2, -1\}\)