Solve for $$ w $$. $$ 4+\frac{3}{w-3}=-\frac{5}{w-4} $$ If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".

Question

UpStudy AI · Accepted Answer

Solve the equation $$ 4+\frac{3}{w-3}=-\frac{5}{w-4} $$.
Solve the equation by following steps:
- step0: Solve for $$w$$:
  $$4+\frac{3}{w-3}=-\frac{5}{w-4}$$
- step1: Find the domain:
  $$4+\frac{3}{w-3}=-\frac{5}{w-4},w \in \left(-\infty,3ight)\cup \left(3,4ight)\cup \left(4,+\inftyight)$$
- step2: Multiply both sides of the equation by LCD:
  $$\left(4+\frac{3}{w-3}ight)\left(w-3ight)\left(w-4ight)=-\frac{5}{w-4}	imes \left(w-3ight)\left(w-4ight)$$
- step3: Simplify the equation:
  $$4w^{2}-25w+36=-5w+15$$
- step4: Move the expression to the left side:
  $$4w^{2}-25w+36-\left(-5w+15ight)=0$$
- step5: Calculate:
  $$4w^{2}-20w+21=0$$
- step6: Factor the expression:
  $$\left(2w-7ight)\left(2w-3ight)=0$$
- step7: Separate into possible cases:
  $$\begin{align}&2w-7=0\&2w-3=0\end{align}$$
- step8: Solve the equation:
  $$\begin{align}&w=\frac{7}{2}\&w=\frac{3}{2}\end{align}$$
- step9: Check if the solution is in the defined range:
  $$\begin{align}&w=\frac{7}{2}\&w=\frac{3}{2}\end{align},w \in \left(-\infty,3ight)\cup \left(3,4ight)\cup \left(4,+\inftyight)$$
- step10: Find the intersection:
  $$\begin{align}&w=\frac{7}{2}\&w=\frac{3}{2}\end{align}$$
- step11: Rewrite:
  $$w_{1}=\frac{3}{2},w_{2}=\frac{7}{2}$$
The solutions to the equation $$4+\frac{3}{w-3}=-\frac{5}{w-4}$$ are $$w = \frac{3}{2}$$ and $$w = \frac{7}{2}$$.
The solutions are $$ w = \frac{3}{2} $$ and $$ w = \frac{7}{2} $$.

Solve for \( w \). \( 4+\frac{3}{w-3}=-\frac{5}{w-4} \) If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".

Real Tutor Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions