Solving Proportions Involving Rational Expressions

Overview

A proportion is an equation that expresses the equality of two ratios or fractions. When solving proportions involving rational expressions, we treat them similarly to solving proportions with simple fractions, but we must be mindful of the rational expressions in the numerators and denominators.

Steps to Solve Proportions Involving Rational Expressions

  1. Write the proportion as two equal fractions.
  2. Cross-multiply the fractions (i.e., multiply the numerator of one fraction by the denominator of the other).
  3. Simplify the equation by solving for the unknown variable.
  4. Check for extraneous solutions by ensuring the denominator is not equal to zero.

Example 1: Solving a Simple Proportion

Consider the following proportion:

\(\frac{1}{x} = \frac{3}{6}\)

Step 1: Cross-multiply the fractions:

\(1 \times 6 = 3 \times x\)

Step 2: Simplify the equation:

\(6 = 3x\)

Step 3: Solve for x:

\(x = \frac{6}{3} = 2\)

Therefore, the solution is x = 2.

Example 2: Solving a Proportion with Rational Expressions

Consider the following proportion:

\(\frac{3}{x+2} = \frac{6}{x-1}\)

Step 1: Cross-multiply the fractions:

\(3 \times (x - 1) = 6 \times (x + 2)\)

Step 2: Expand both sides:

\(3x - 3 = 6x + 12\)

Step 3: Simplify the equation:

\(3x - 6x = 12 + 3\)

\(-3x = 15\)

Step 4: Solve for x:

\(x = \frac{15}{-3} = -5\)

Therefore, the solution is x = -5.

Practice Questions

  1. Question 1: Solve the proportion:

    \(\frac{4}{x+3} = \frac{8}{x-2}\)

    Solution

    Step 1: Cross-multiply the fractions:

    \(4 \times (x - 2) = 8 \times (x + 3)\)

    Step 2: Expand both sides:

    \(4x - 8 = 8x + 24\)

    Step 3: Simplify the equation:

    \(4x - 8x = 24 + 8\)

    \(-4x = 32\)

    Step 4: Solve for x:

    \(x = \frac{32}{-4} = -8\)

    Therefore, the solution is x = -8.

  2. Question 2: Solve the proportion:

    \(\frac{x}{x+5} = \frac{4}{x+1}\)

    Solution

    Step 1: Cross-multiply the fractions:

    \(x \times (x + 1) = 4 \times (x + 5)\)

    Step 2: Expand both sides:

    \(x^2 + x = 4x + 20\)

    Step 3: Rearrange the equation:

    \(x^2 + x - 4x - 20 = 0\)

    \(x^2 - 3x - 20 = 0\)

    Step 4: Factor the quadratic equation:

    \((x - 5)(x + 4) = 0\)

    Step 5: Solve for x:

    \(x = 5 \quad \text{or} \quad x = -4\)

    Therefore, the solutions are x = 5 and x = -4.

  3. Question 3: Solve the proportion:

    \(\frac{x+3}{x-1} = \frac{2x}{x+4}\)

    Solution

    Step 1: Cross-multiply the fractions:

    \((x + 3) \times (x + 4) = 2x \times (x - 1)\)

    Step 2: Expand both sides:

    \(x^2 + 7x + 12 = 2x^2 - 2x\)

    Step 3: Rearrange the equation:

    \(x^2 + 7x + 12 - 2x^2 + 2x = 0\)

    \(-x^2 + 9x + 12 = 0\)

    Step 4: Multiply both sides by -1 to make the leading coefficient positive:

    \(x^2 - 9x - 12 = 0\)

    Step 5: Solve using the quadratic formula:

    \(x = \frac{-9 \pm \sqrt{9^2 - 4(1)(-12)}}{2(1)}\)

    \(x = \frac{-9 \pm \sqrt{81 + 48}}{2}\)

    \(x = \frac{-9 \pm \sqrt{129}}{2}\)

    Therefore, the solutions are approximately x ≈ -11.18 and x ≈ 2.18.

  4. Question 4: Solve the proportion:

    \(\frac{2x+5}{x-3} = \frac{x-1}{x+2}\)

    Solution

    Step 1: Cross-multiply the fractions:

    \((2x + 5) \times (x + 2) = (x - 1) \times (x - 3)\)

    Step 2: Expand both sides:

    \(2x^2 + 4x + 5x + 10 = x^2 - 3x - x + 3\)

    Step 3: Simplify the equation:

    \(2x^2 + 9x + 10 = x^2 - 4x + 3\)

    Step 4: Rearrange the equation:

    \(2x^2 + 9x + 10 - x^2 + 4x - 3 = 0\)

    \(x^2 + 13x + 7 = 0\)

    Step 5: Solve using the quadratic formula:

    \(x = \frac{-13 \pm \sqrt{13^2 - 4(1)(7)}}{2(1)}\)

    \(x = \frac{-13 \pm \sqrt{169 - 28}}{2}\)

    \(x = \frac{-13 \pm \sqrt{141}}{2}\)

    Therefore, the solutions are approximately x ≈ -13.46 and x ≈ 0.46.

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