Question: Solve for $ x $: $ \frac{2}{3}(x + 4) = \frac{5}{6}(x - 1) $.

["Solving Linear Equations: A Step-by-Step Guide to $ \frac{2}{3}(x + 4) = \frac{5}{6}(x - 1) $", "Solving linear equations is a foundational skill in algebra that appears frequently in school curricula and real-world problem-solving. One common challenge for students is equations involving fractions and parentheses. In this article, we’ll explore how to solve the equation:", "$$\n\frac{2}{3}(x + 4) = \frac{5}{6}(x - 1)\n$$", "This problem demonstrates how to isolate the variable $ x $ while carefully handling fractions and distributing terms. By following a clear, step-by-step approach, anyone can master solving equations like this with confidence.", "---", "### Step 1: Eliminate Fractions by Finding a Common Denominator", "The left side has denominator 3 and the right side has denominator 6. The smallest common denominator is 6. To simplify, multiply both sides of the equation by 6:", "$$\n6 \cdot \left[\frac{2}{3}(x + 4)\right] = 6 \cdot \left[\frac{5}{6}(x - 1)\right]\n$$", "Using the distributive property:", "$$\n6 \cdot \frac{2}{3}(x + 4) = 5(x - 1)\n$$", "Simplify:", "$$\n\frac{12}{3}(x + 4) = 5(x - 1)\n\quad \Rightarrow \quad\n4(x + 4) = 5(x - 1)\n$$", "---", "### Step 2: Expand Both Sides of the Equation", "Now expand both sides:", "Left side:\n$$\n4(x + 4) = 4x + 16\n$$", "Right side:\n$$\n5(x - 1) = 5x - 5\n$$", "So the equation becomes:", "$$\n4x + 16 = 5x - 5\n$$", "---", "### Step 3: Collect Like Terms (Isolate $ x $)", "Our goal is to get all the $ x $-terms on one side and constant terms on the other. Subtract $ 4x $ from both sides:", "$$\n16 = x - 5\n$$", "Next, add 5 to both sides to solve for $ x $:", "$$\n16 + 5 = x\n\quad \Rightarrow \quad\nx = 21\n$$", "---", "### Step 4: Check for Extraneous Solutions", "Although rare in linear equations, it’s always good practice to substitute the solution back into the original equation to verify correctness.", "Substitute $ x = 21 $:", "Left side:\n$$\n\frac{2}{3}(21 + 4) = \frac{2}{3}(25) = \frac{50}{3}\n$$", "Right side:\n$$\n\frac{5}{6}(21 - 1) = \frac{5}{6}(20) = \frac{100}{6} = \frac{50}{3}\n$$", "Both sides are equal, confirming the solution is valid.", "---", "### Summary", "- The equation $ \frac{2}{3}(x + 4) = \frac{5}{6}(x - 1) $ was solved by eliminating fractions using a common denominator.\n- Expansion and simple algebra allowed us to isolate $ x $.\n- A solution check confirmed correctness.", "Mastering such equations strengthens algebraic reasoning, which is essential for advanced math and STEM applications. If you're struggling with equations involving fractions, practice with different coefficients and parentheses—and always verify your work!", "---", "Key Takeaways:", "- Use the Least Common Denominator (LCD) to eliminate fractions efficiently.\n- Apply the distributive property carefully.\n- Combine like terms and isolate $ x $ step by step.\n- Always substitute back to verify solutions.", "By following these clear steps, solving equations like this becomes manageable and even intuitive. Keep practicing, and watch your algebra confidence grow!"]