2x + 3\left(-\frac{32}{11}\right) = -6 \\

["# Solving the Equation: 2x + 3\left(-\frac{32}{11}\right) = -6", "When solving linear equations, one common challenge is properly handling negative fractions and simplifying expressions before isolating the variable. This article guides you through solving the equation:", "$$\n2x + 3\left(-\frac{32}{11}\right) = -6\n$$", "by step-by-step clarification, focusing on simplifying constants and isolating $ x $.", "---", "## Step 1: Simplify the Constant Term", "The equation contains a term with a negative fraction:", "$$\n3\left(-\frac{32}{11}\right) = -\frac{96}{11}\n$$", "So the equation becomes:", "$$\n2x - \frac{96}{11} = -6\n$$", "Why simplify?\nSimplifying expressions before moving terms ensures accuracy and reduces complexity when isolating $ x $. Fraction arithmetic is crucial here—never forget the negative sign.", "---", "## Step 2: Add $\frac{96}{11}$ to Both Sides", "To isolate $ 2x $, add $ \frac{96}{11} $ to both sides:", "$$\n2x = -6 + \frac{96}{11}\n$$", "Now convert $ -6 $ into a fraction with denominator 11:", "$$\n-6 = -\frac{66}{11}\n$$", "Add the fractions:", "$$\n2x = -\frac{66}{11} + \frac{96}{11} = \frac{30}{11}\n$$", "---", "## Step 3: Divide Both Sides by 2", "Now divide both sides by 2 to solve for $ x $:", "$$\nx = \frac{30}{11} \div 2 = \frac{30}{11} \cdot \frac{1}{2} = \frac{30}{22} = \frac{15}{11}\n$$", "---", "## Final Answer", "$$\nx = \frac{15}{11}\n$$", "---", "## Why This Equation Matters", "Understanding how to manipulate constants, especially those involving negative fractions, is essential in algebra. Mastering this skill helps solve more complex equations in science, engineering, and economics—fields where precise calculations are critical.", "---", "### Bonus: Check Your Work", "Plug $ x = \frac{15}{11} $ back into the original equation:", "Left-hand side:", "$$\n2\left(\frac{15}{11}\right) + 3\left(-\frac{32}{11}\right) = \frac{30}{11} - \frac{96}{11} = -\frac{66}{11} = -6\n$$", "Matches the right-hand side, confirming the solution is correct.", "---", "### Key Topics Covered", "- Distributing negative signs\n- Adding/subtracting fractions with common and different denominators\n- Isolating variables in linear equations\n- Verifying solutions", "Mastering these concepts puts you on the right path to confidently solving algebra equations."]