Question

what are the solutions to the equation \\(\frac{w}{2w - 3}=\frac{4}{w}\\)?
\\(w = -6\\) and \\(w = -2\\)
\\(w = 0\\), \\(w = 2\\), and \\(w = 6\\)
\\(w = 0\\) and \\(w = \frac{3}{2}\\)
\\(w = 2\\) and \\(w = 6\\)

Explanation:

Step1: Note undefined values

First, identify values that make denominators zero: $2w-3=0 \implies w=\frac{3}{2}$; $w=0$. These cannot be solutions.

Step2: Cross-multiply to eliminate fractions

Multiply both sides by $w(2w-3)$:
$$w \cdot w = 4(2w-3)$$

Step3: Simplify to quadratic equation

Expand and rearrange terms:
$$w^2 = 8w - 12$$
$$w^2 - 8w + 12 = 0$$

Step4: Factor the quadratic

Factor the polynomial:
$$(w-2)(w-6)=0$$

Step5: Solve for w

Set each factor equal to zero:
$w-2=0 \implies w=2$; $w-6=0 \implies w=6$
Both values are valid (not undefined).

Answer:

w = 2 and w = 6