\( \frac{1}{5} + \frac{1}{x} = \frac{11}{30} \).
["Solving the Equation ( \frac{1}{5} + \frac{1}{x} = \frac{11}{30} ): Step-by-Step Guide", "Mathematics often presents us with insightful equations that challenge our understanding and sharpen problem-solving skills. One such equation is:", "[\n\frac{1}{5} + \frac{1}{x} = \frac{11}{30}\n]", "In this article, we’ll walk through solving this equation step-by-step, explaining each move in detail. Whether you're a student tackling algebra or someone reviewing foundational algebra skills, this guide will help you confidently solve rational equations like this one.", "---", "### Step 1: Isolate the Variable Term", "Begin by isolating the term containing ( x ). Subtract ( \frac{1}{5} ) from both sides of the equation:", "[\n\frac{1}{x} = \frac{11}{30} - \frac{1}{5}\n]", "Before performing the subtraction, find a common denominator for the fractions on the right-hand side. The least common denominator of 30 and 5 is 30.", "[\n\frac{1}{5} = \frac{6}{30}\n]", "Now rewrite the equation:", "[\n\frac{1}{x} = \frac{11}{30} - \frac{6}{30} = \frac{5}{30} = \frac{1}{6}\n]", "So now the equation simplifies to:", "[\n\frac{1}{x} = \frac{1}{6}\n]", "---", "### Step 2: Solve for ( x )", "Since ( \frac{1}{x} = \frac{1}{6} ), you recognize that the reciprocal relationship gives:", "[\nx = 6\n]", "This is the solution to the equation.", "---", "### Step 3: Check Your Solution (Important!)", "To ensure accuracy, substitute ( x = 6 ) back into the original equation:", "[\n\frac{1}{5} + \frac{1}{6} = \frac{6 + 5}{30} = \frac{11}{30}\n]", "The left-hand side matches the right-hand side exactly, confirming that ( x = 6 ) is indeed correct.", "---", "### Why This Equation Matters", "Solving equations with rational expressions like this one builds core algebraic thinking:\n- Combining fractions\n- Working with denominators and common denominators\n- Manipulating both sides of an equation\n- Understanding reciprocals", "These skills are fundamental for more advanced math topics, including calculus, physics, and engineering applications.", "---", "### Final Answer", "[\n\boxed{x = 6}\n]", "---", "### Additional Tips", "- Always simplify fractions before proceeding.\n- Check for extraneous solutions—some algebraic steps might introduce false answers.\n- Practice similar equations to strengthen fluency.\n- Use online equation solvers as a learning aid, but aim to solve them manually for deeper understanding.", "---", "Keywords: solve rational equation, algebra problems, step-by-step solution, ( \frac{1}{x} = \frac{11}{30} - \frac{1}{5} ), equation solving technique, algebraic equations, math tutorial", "---", "Eager to sharpen your algebra skills? Mastering these equations opens doors to higher-level math and real-world problem solving. Start practicing today!"]
