Solve by graphing: $$ \frac{3}{2 x-1}=\frac{x+2}{x} $$ $$ \left\{\begin{array}{l}\{-1,0,1\} \ \{1\} \ \{-2,3\} \ \{-1,1\}\end{array} ight. $$

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Solve by graphing: $$ \frac{3}{2 x-1}=\frac{x+2}{x} $$ $$ \left\{\begin{array}{l}\{-1,0,1\} \ \{1\} \ \{-2,3\} \ \{-1,1\}\end{array}ight. $$

UpStudy AI · Accepted Answer

Solve the equation $$ \frac{3}{2x-1}=\frac{x+2}{x} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\frac{3}{2x-1}=\frac{x+2}{x}$$
- step1: Find the domain:
  $$\frac{3}{2x-1}=\frac{x+2}{x},x \in \left(-\infty,0ight)\cup \left(0,\frac{1}{2}ight)\cup \left(\frac{1}{2},+\inftyight)$$
- step2: Cross multiply:
  $$3x=\left(2x-1ight)\left(x+2ight)$$
- step3: Expand the expression:
  $$3x=2x^{2}+3x-2$$
- step4: Cancel equal terms:
  $$0=2x^{2}-2$$
- step5: Swap the sides:
  $$2x^{2}-2=0$$
- step6: Move the constant to the right side:
  $$2x^{2}=2$$
- step7: Divide both sides:
  $$\frac{2x^{2}}{2}=\frac{2}{2}$$
- step8: Divide the numbers:
  $$x^{2}=1$$
- step9: Simplify the expression:
  $$x=\pm \sqrt{1}$$
- step10: Simplify:
  $$x=\pm 1$$
- step11: Separate into possible cases:
  $$\begin{align}&x=1\&x=-1\end{align}$$
- step12: Check if the solution is in the defined range:
  $$\begin{align}&x=1\&x=-1\end{align},x \in \left(-\infty,0ight)\cup \left(0,\frac{1}{2}ight)\cup \left(\frac{1}{2},+\inftyight)$$
- step13: Find the intersection:
  $$\begin{align}&x=1\&x=-1\end{align}$$
- step14: Rewrite:
  $$x_{1}=-1,x_{2}=1$$
The solutions to the equation $$ \frac{3}{2x-1}=\frac{x+2}{x} $$ are $$ x = -1 $$ and $$ x = 1 $$.

Therefore, the solutions are $$ \{-1,1\} $$.
The solutions are $$ x = -1 $$ and $$ x = 1 $$.

Solve by graphing:

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