Solving Rational Equations Step-by-Step Guide

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This article provides a comprehensive guide to solving rational equations and inequalities. We will walk through several examples, providing step-by-step solutions and explanations. Understanding how to solve these types of problems is crucial for various mathematical applications, particularly in algebra and calculus. Mastering these techniques will empower you to tackle complex problems with confidence. This guide aims to equip you with the skills and knowledge necessary to solve rational equations and inequalities efficiently and accurately. Let's dive into the methods and strategies required to solve these types of mathematical challenges.

1. Solving the Equation: (2x-3)/x = 1/(x+1)

To solve the equation (2x-3)/x = 1/(x+1), our primary goal is to eliminate the fractions. We achieve this by multiplying both sides of the equation by the least common denominator (LCD). In this case, the LCD is x(x+1). Multiplying by the LCD allows us to clear the fractions, simplifying the equation into a more manageable form. This technique is fundamental in solving rational equations and is a crucial step towards finding the solution. Let's explore this process step-by-step to ensure a clear understanding.

Step-by-Step Solution

  1. Identify the Least Common Denominator (LCD):

    • The denominators in the equation are x and (x+1). Therefore, the LCD is x(x+1).

  2. Multiply both sides of the equation by the LCD:

    • (2x-3)/x * x(x+1) = 1/(x+1) * x(x+1)
    • This simplifies to (2x-3)(x+1) = x

  3. Expand and simplify the equation:

    • Expanding the left side, we get 2x^2 + 2x - 3x - 3 = x
    • Combining like terms, the equation becomes 2x^2 - x - 3 = x

  4. Move all terms to one side to set the equation to zero:

    • Subtract x from both sides: 2x^2 - x - 3 - x = 0
    • This simplifies to 2x^2 - 2x - 3 = 0

  5. Solve the quadratic equation:

    • We can use the quadratic formula to solve for x: x = [-b ± √(b^2 - 4ac)] / (2a)
    • In this case, a = 2, b = -2, and c = -3
    • Plugging in these values, we get x = [2 ± √((-2)^2 - 4 * 2 * -3)] / (2 * 2)
    • Simplifying further, x = [2 ± √(4 + 24)] / 4
    • x = [2 ± √28] / 4
    • x = [2 ± 2√7] / 4
    • x = (1 ± √7) / 2

  6. Check for extraneous solutions:

    • Extraneous solutions are solutions that satisfy the transformed equation but not the original equation. This often happens when dealing with rational equations due to potential division by zero.
    • We need to ensure that our solutions do not make the denominators in the original equation equal to zero. The original denominators were x and (x+1).
    • If x = 0 or x = -1, the denominators would be zero, making the solutions invalid.

  7. Verify the solutions:

    • The solutions we found are x = (1 + √7) / 2 and x = (1 - √7) / 2. Neither of these values makes the denominators zero, so they are both valid solutions.

Final Answer

The solutions to the equation (2x-3)/x = 1/(x+1) are x = (1 + √7) / 2 and x = (1 - √7) / 2. These values satisfy the original equation without causing any division by zero, confirming their validity. This careful approach ensures we have accurate solutions.

2. Solving the Equation: (x+7)/(3x^2-2x-1) = 1

To solve the equation (x+7)/(3x^2-2x-1) = 1, we first need to clear the fraction. Clearing the fraction involves multiplying both sides of the equation by the denominator, which in this case is 3x^2-2x-1. This step is crucial for simplifying the equation and making it easier to solve. By eliminating the denominator, we convert the rational equation into a more manageable form, typically a polynomial equation. This allows us to apply standard algebraic techniques to find the solutions. Let's proceed with this step-by-step approach to solve the equation.

Step-by-Step Solution

  1. Multiply both sides by the denominator:

    • Multiply both sides of the equation by (3x^2 - 2x - 1): (x+7) = 1 * (3x^2 - 2x - 1)
    • This simplifies to x + 7 = 3x^2 - 2x - 1

  2. Rearrange the equation into a standard quadratic form:

    • Move all terms to one side to set the equation to zero: 0 = 3x^2 - 2x - 1 - x - 7
    • Combine like terms: 0 = 3x^2 - 3x - 8

  3. Solve the quadratic equation:

    • The quadratic equation is in the form ax^2 + bx + c = 0, where a = 3, b = -3, and c = -8
    • Use the quadratic formula: x = [-b ± √(b^2 - 4ac)] / (2a)
    • Substitute the values: x = [3 ± √((-3)^2 - 4 * 3 * -8)] / (2 * 3)
    • Simplify: x = [3 ± √(9 + 96)] / 6
    • x = [3 ± √105] / 6

  4. Check for extraneous solutions:

    • Extraneous solutions occur when a solution makes the denominator of the original equation equal to zero.
    • The original denominator was 3x^2 - 2x - 1. We need to find the values of x that make this expression zero.
    • Factor the quadratic: 3x^2 - 2x - 1 = (3x + 1)(x - 1)
    • Set each factor to zero: 3x + 1 = 0 or x - 1 = 0
    • Solve for x: x = -1/3 or x = 1
    • These values would make the denominator zero, so they are not valid solutions.

  5. Verify the solutions:

    • The solutions we found are x = (3 + √105) / 6 and x = (3 - √105) / 6. We need to check if these values are extraneous.
    • Since these values are not equal to -1/3 or 1, they are not extraneous solutions.

Final Answer

The solutions to the equation (x+7)/(3x^2-2x-1) = 1 are x = (3 + √105) / 6 and x = (3 - √105) / 6. These values satisfy the original equation without causing the denominator to be zero, confirming their validity. This careful check is essential for accuracy.

3. Solving the Equation: h(x) = (x-1)/(2x+7) - 8

To solve the equation h(x) = (x-1)/(2x+7) - 8, we need to find the values of x for which this equation holds true. The first step involves isolating the rational expression. Isolating the rational expression often simplifies the process of solving rational equations. This involves moving all other terms to the opposite side of the equation, making it easier to work with the fractional part. By doing so, we can then proceed to clear the fraction and solve for x. Let's proceed step by step.

Step-by-Step Solution

  1. Set h(x) equal to zero:

    • To solve for x, we set h(x) = 0: 0 = (x-1)/(2x+7) - 8

  2. Isolate the rational expression:

    • Add 8 to both sides: 8 = (x-1)/(2x+7)

  3. Clear the fraction by multiplying both sides by the denominator:

    • Multiply both sides by (2x+7): 8(2x+7) = x - 1
    • Distribute on the left side: 16x + 56 = x - 1

  4. Move all terms involving x to one side and constants to the other side:

    • Subtract x from both sides: 16x - x + 56 = -1
    • Subtract 56 from both sides: 15x = -1 - 56
    • Simplify: 15x = -57

  5. Solve for x:

    • Divide both sides by 15: x = -57 / 15
    • Simplify the fraction: x = -19 / 5

  6. Check for extraneous solutions:

    • The denominator in the original equation is (2x+7). We need to ensure that this is not equal to zero for our solution.
    • Set 2x + 7 = 0 and solve for x: 2x = -7, x = -7/2
    • Since our solution x = -19/5 is not equal to -7/2, it is not an extraneous solution.

Final Answer

The solution to the equation h(x) = (x-1)/(2x+7) - 8 is x = -19/5. This value satisfies the equation without making the denominator zero, confirming its validity. Double-checking for extraneous solutions is crucial in rational equations.

4. Solving the Equation: y/(3x) = 4/(x^2-6)

To solve the equation y/(3x) = 4/(x^2-6), we first need to clear the fractions. Clearing fractions is a fundamental step in solving rational equations. This involves multiplying both sides of the equation by the least common denominator (LCD). In this case, the LCD is 3x(x^2-6). Multiplying by the LCD eliminates the fractions, transforming the equation into a more manageable form that we can solve using standard algebraic techniques. This process is critical for simplifying the equation and finding the solutions. Let’s proceed step by step to ensure a clear understanding.

Step-by-Step Solution

  1. Identify the Least Common Denominator (LCD):

    • The denominators are 3x and (x^2 - 6), so the LCD is 3x(x^2 - 6).

  2. Multiply both sides of the equation by the LCD:

    • [y/(3x)] * [3x(x^2 - 6)] = [4/(x^2 - 6)] * [3x(x^2 - 6)]
    • This simplifies to y(x^2 - 6) = 12x

  3. Expand and rearrange the equation:

    • Expanding the left side, we get yx^2 - 6y = 12x
    • Rearrange the equation to get a quadratic form in terms of x: yx^2 - 12x - 6y = 0

  4. Solve for x using the quadratic formula:

    • The quadratic formula is x = [-b ± √(b^2 - 4ac)] / (2a)
    • Here, a = y, b = -12, and c = -6y
    • Plugging in the values, we get x = [12 ± √((-12)^2 - 4 * y * (-6y))] / (2y)
    • Simplifying, x = [12 ± √(144 + 24y^2)] / (2y)
    • x = [12 ± √(24(6 + y^2))] / (2y)
    • x = [12 ± 2√(6(6 + y^2))] / (2y)
    • x = [6 ± √(6(6 + y^2))] / y

  5. Check for extraneous solutions:

    • The original denominators were 3x and (x^2 - 6). We need to ensure that these are not equal to zero.
    • 3x ≠ 0 implies x ≠ 0
    • x^2 - 6 ≠ 0 implies x ≠ ±√6
    • We need to ensure that the solutions we found do not violate these conditions.

Final Answer

The solution to the equation y/(3x) = 4/(x^2-6) is x = [6 ± √(6(6 + y^2))] / y, provided that x ≠ 0 and x ≠ ±√6. This value satisfies the equation without making the denominator zero, confirming its validity. Verifying solutions against the original equation is crucial for accuracy.

In this comprehensive guide, we've explored the methods for solving various rational equations. Each solution requires careful attention to detail, especially when checking for extraneous solutions. Mastering these techniques is crucial for success in algebra and beyond. By understanding the underlying principles and practicing diligently, you can confidently tackle any rational equation that comes your way. Remember to always check your solutions and ensure they are valid within the original context of the problem. With consistent practice, solving rational equations will become a straightforward and manageable task. Keep practicing, and you'll master these skills in no time!