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Solve for all values of xx. x−2x−3=2x - \frac{2}{x - 3} = 2

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Full solution

Q. Solve for all values of xx. x−2x−3=2x - \frac{2}{x - 3} = 2
  1. Multiply and Simplify: First, let's get rid of the fraction by multiplying everything by (x−3)(x - 3).(x−2x−3)(x−3)=2(x−3)(x - \frac{2}{x - 3})(x - 3) = 2(x - 3)
  2. Distribute and Expand: Now distribute x−3x - 3 on the left side and simplify the right side.x(x−3)−2=2x−6x(x - 3) - 2 = 2x - 6
  3. Subtract and Simplify: Expand the left side of the equation. x2−3x−2=2x−6x^2 - 3x - 2 = 2x - 6
  4. Add and Set to Zero: Subtract 2x2x from both sides to get a quadratic equation.\newlinex2−5x−2=−6x^2 - 5x - 2 = -6
  5. Factor the Equation: Add 66 to both sides to set the quadratic equation to zero.\newlinex2−5x+4=0x^2 - 5x + 4 = 0
  6. Set Factors Equal: Factor the quadratic equation.\newline(x−4)(x−1)=0(x - 4)(x - 1) = 0
  7. Solve for x: Set each factor equal to zero and solve for x.\newlinex−4=0x - 4 = 0 or x−1=0x - 1 = 0
  8. Solve for x: Solve the first equation for x.\newlinex=4x = 4
  9. Solve for x: Solve the first equation for x.\newlinex=4x = 4Solve the second equation for x.\newlinex=1x = 1

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