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Question

Difficulty: HardRadical and Rational Equations

Consider the equation:

x+3x−1+4x=4x2−x\frac{x + 3}{x - 1} + \frac{4}{x} = \frac{4}{x^2 - x}

What is the value of x+3x + 3 for the real value of xx that satisfies this equation?

  1. -5Answer
  2. B
    4
  3. C
    11
  4. D
    2

Answer

-5
The correct answer is −5-5. Multiplying the rational equation by the common denominator x(x−1)x(x - 1) gives the simplified equation x(x+3)+4(x−1)=4x(x + 3) + 4(x - 1) = 4. Expanding and rearranging the terms leads to the quadratic equation x2+7x−8=0x^2 + 7x - 8 = 0. Factoring this quadratic yields (x+8)(x−1)=0(x + 8)(x - 1) = 0, giving the potential solutions x=−8x = -8 and x=1x = 1. Substituting these values back into the original equation shows that x=1x = 1 causes division by zero, making it extraneous. Thus, the only valid solution is x=−8x = -8. Substituting this valid solution into the expression x+3x + 3 yields −8+3=−5-8 + 3 = -5.

Step-by-Step Solution

1
Find the common denominator of the rational terms.
The denominators are x−1x - 1, xx, and x2−xx^2 - x. The common denominator is x(x−1)=x2−xx(x - 1) = x^2 - x.
Finding a common denominator allows us to eliminate the fractions by multiplying both sides.
2
Multiply the entire equation by the common denominator x(x−1)x(x - 1) to clear the denominators.
x(x+3)+4(x−1)=4x(x + 3) + 4(x - 1) = 4
This simplifies the rational equation into a polynomial equation.
3
Expand and simplify the resulting equation.
x2+3x+4x−4=4x^2 + 3x + 4x - 4 = 4, which simplifies to x2+7x−4=4x^2 + 7x - 4 = 4.
Expanding the terms allows us to collect like terms and identify the type of polynomial equation.
4
Write the quadratic equation in standard form and factor it.
x2+7x−8=0x^2 + 7x - 8 = 0, which factors as (x+8)(x−1)=0(x + 8)(x - 1) = 0.
Setting the quadratic equation to zero allows us to find its roots by factoring.
5
Find the potential solutions and check for extraneous values in the original equation.
The potential solutions are x=−8x = -8 and x=1x = 1. The value x=1x = 1 is extraneous because it makes the denominators x−1x - 1 and x2−xx^2 - x equal to zero. The only valid solution is x=−8x = -8.
We must check if the roots make any denominator in the original equation zero, as division by zero is undefined.
6
Evaluate the expression x+3x + 3 using the valid solution.
−8+3=−5-8 + 3 = -5
The question asks for the value of the expression x+3x + 3, not the value of xx itself.

Key Concept

Solving rational equations and checking for extraneous solutions
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Topics

Radical and Rational Equations

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