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

Explanation:

Step1: Cross - multiply

Cross - multiplying the equation $\frac{w}{2w - 3}=\frac{4}{w}$ gives $w\times w=4\times(2w - 3)$. So, $w^{2}=8w - 12$.

Step2: Rearrange to quadratic form

Rearrange the equation to the standard quadratic form $ax^{2}+bx + c = 0$. $w^{2}-8w + 12=0$.

Step3: Factor the quadratic

Factor the quadratic equation: $w^{2}-8w + 12=(w - 2)(w - 6)=0$.

Step4: Solve for w

Set each factor equal to zero: If $w - 2=0$, then $w = 2$; if $w - 6=0$, then $w = 6$. We also need to check for extraneous solutions. The original equation $\frac{w}{2w - 3}=\frac{4}{w}$ has a denominator of $2w-3$ and $w$. When $w = 0$, the right - hand side $\frac{4}{w}$ is undefined, and when $w=\frac{3}{2}$, the left - hand side $\frac{w}{2w - 3}$ is undefined. So the valid solutions are $w = 2$ and $w = 6$.

Answer:

$w = 2$ and $w = 6$