Question: Solve for $ x $: $ \frac{2}{x - 1} + \frac{3}{x + 2} = 1 $.

February 22, 2026

$Question: Solve for $ x $: $ \frac{2}{x - 1} + \frac{3}{x + 2} = 1 $.$

["Solve for $ x $: A Step-by-Step Guide to Solving the Equation $ \frac{2}{x - 1} + \frac{3}{x + 2} = 1 $", "Solving rational equations like $ \frac{2}{x - 1} + \frac{3}{x + 2} = 1 $ can seem challenging at first, but with a clear step-by-step approach, you can solve for $ x $ efficiently. This article walks you through solving the equation $ \frac{2}{x - 1} + \frac{3}{x + 2} = 1 $ while highlighting key algebraic techniques and common pitfalls to avoid.", "---", "### Step 1: Identify Restrictions", "Before solving, determine where the expression is undefined. The denominators $ x - 1 $ and $ x + 2 $ cannot be zero:", "$$\nx - 1 \neq 0 \Rightarrow x \neq 1 \\nx + 2 \neq 0 \Rightarrow x \neq -2\n$$", "These values must be excluded from the solution set.", "---", "### Step 2: Find the Common Denominator", "The denominators are $ x - 1 $ and $ x + 2 $. The least common denominator (LCD) is:", "$$\n(x - 1)(x + 2)\n$$", "Multiply both sides of the equation by the LCD to eliminate the fractions:", "$$\n(x - 1)(x + 2) \left( \frac{2}{x - 1} + \frac{3}{x + 2} \right) = (x - 1)(x + 2) \cdot 1\n$$", "---", "### Step 3: Distribute and Simplify", "Expand both sides:", "$$\n2(x + 2) + 3(x - 1) = (x - 1)(x + 2)\n$$", "Now simplify each term:", "Left-hand side:", "$$\n2x + 4 + 3x - 3 = 5x + 1\n$$", "Right-hand side:", "$$\n(x - 1)(x + 2) = x^2 + 2x - x - 2 = x^2 + x - 2\n$$", "So the equation becomes:", "$$\n5x + 1 = x^2 + x - 2\n$$", "---", "### Step 4: Rearrange into Standard Quadratic Form", "Move all terms to one side:", "$$\n0 = x^2 + x - 2 - 5x - 1\n\Rightarrow x^2 - 4x - 3 = 0\n$$", "Now solve the quadratic equation:", "$$\nx^2 - 4x - 3 = 0\n$$", "Use the quadratic formula:", "$$\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-3)}}{2(1)} = \frac{4 \pm \sqrt{16 + 12}}{2} = \frac{4 \pm \sqrt{28}}{2}\n$$", "Simplify $ \sqrt{28} = 2\sqrt{7} $:", "$$\nx = \frac{4 \pm 2\sqrt{7}}{2} = 2 \pm \sqrt{7}\n$$", "---", "### Step 5: Check for Extraneous Solutions", "Recall the restrictions: $ x \neq 1 $ and $ x \neq -2 $. Neither $ 2 + \sqrt{7} $ nor $ 2 - \sqrt{7} $ equals these excluded values, so both are valid.", "Approximating:", "$$\n\sqrt{7} \approx 2.6458 \Rightarrow x \approx 4.6458 \quad \ ext{and} \quad x \approx -0.6458\n$$", "---", "### Final Answer", "$$\n\boxed{x = 2 + \sqrt{7} \quad \ ext{or} \quad x = 2 - \sqrt{7}}\n$$", "---", "### Why This Matters for Students and Learners", "Mastering rational equations builds problem-solving skills essential in algebra and beyond. By carefully eliminating denominators, simplifying, and verifying solutions, you avoid common mistakes and strengthen your mathematical foundation. Whether you're learning algebra for school or self-study, this step-by-step method ensures accuracy and confidence.", "---", "Keywords: solve for $ x $, rational equations, solve $ \frac{2}{x - 1} + \frac{3}{x + 2} = 1 $, quadratic equation, algebraic methods, step-by-step solving, excluded values, common denominator, quadratic formula, $ x = 2 \pm \sqrt{7} $", "Meta Description: Step-by-step guide to solving $ \frac{2}{x - 1} + \frac{3}{x + 2} = 1 $. Learn how to eliminate denominators, simplify, and check for valid solutions — perfect for students and algebra learners."]

Question: Solve for $ x $: $ \frac{2}{x - 1} + \frac{3}{x + 2} = 1 $.

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