Verified Answer :

  • Identify domain constraints \(x eq -1\) and \(x eq 2\) for the given rational equation.
  • Multiply the entire equation by the common denominator \((x+1)(x-2)\) to remove fractions.
  • Simplify the resulting equation to \(3x^2 - 2x + 4 = 3x^2 - 3x - 6\), which reduces to the linear equation \(x = -10\).
  • Verify the solution against the domain and conclude that \(\boxed{-10}\).

Explanation

  1. Problem Analysis and Domain Constraints

We are given the rational equation:
\[\frac{3x}{x+1} + \frac{4}{x-2} = 3.\]
First, we note that the expression is undefined when the denominators are zero. Thus, we must have \(x + 1 eq 0\) and \(x - 2 eq 0\), which means \(x eq -1\) and \(x eq 2\). We aim to solve this equation for \(x\) by transforming it into a more manageable linear or quadratic equation.

  1. Eliminating the Denominators

To eliminate the fractions, we multiply the entire equation by the common denominator, which is \((x+1)(x-2)\):
\[(x+1)(x-2) \left( \frac{3x}{x+1} + \frac{4}{x-2} \right) = 3(x+1)(x-2).\]
Distributing this product gives:
\[3x(x-2) + 4(x+1) = 3(x^2 - 2x + x - 2).\]
Expanding the left side and simplifying the expression inside the parentheses on the right side:
\[3x^2 - 6x + 4x + 4 = 3(x^2 - x - 2).\]
\[3x^2 - 2x + 4 = 3x^2 - 3x - 6.\]

  1. Solving the Equation

Now, we isolate the terms involving \(x\) to solve for it:
\[3x^2 - 2x + 4 = 3x^2 - 3x - 6.\]
Subtracting \(3x^2\) from both sides of the equation simplifies it to a linear equation:
\[-2x + 4 = -3x - 6.\]
Adding \(3x\) to both sides:
\[x + 4 = -6.\]
Subtracting 4 from both sides:
\[x = -10.\]

  1. Validation and Final Answer

We must verify that our solution \(x = -10\) satisfies the domain constraints established in Step 1. Since \(x = -10 eq -1\) and \(x = -10 eq 2\), the solution is valid. We can check this by substituting \(x = -10\) back into the original equation:
\[\frac{3(-10)}{-10+1} + \frac{4}{-10-2} = \frac{-30}{-9} + \frac{4}{-12} = \frac{10}{3} - \frac{1}{3} = \frac{9}{3} = 3.\]
The solution holds true, confirming the value of \(x\). Therefore, the solution is \(\boxed{-10}\).

Examples

This type of equation is commonly found when modeling rate problems. For example, if two workers are cleaning a space and together they take a specific amount of time to finish, their individual rates might be represented as parts of a rational expression. Finding the value of \(x\) allows us to determine the time or rate at which one of them works compared to the other. Mastering these techniques helps in efficiency planning, logistics, and engineering projects.

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