The equation is \( \frac{4 + x}{10 + x} = 0.6 \).

["# Solving the Equation: ( \frac{4 + x}{10 + x} = 0.6 ) — A Step-by-Step Guide", "Understanding how to solve equations is fundamental in algebra, and one commonly encountered problem is solving rational equations like ( \frac{4 + x}{10 + x} = 0.6 ). In this article, we will break down the steps to solve this equation clearly and efficiently, explain its real-world relevance, and highlight key math concepts. If you’re curious about how to simplify and solve linear equations involving fractions, especially with decimal values, this guide is for you.", "---", "## What Is the Equation?", "The equation to solve is:", "[\n\frac{4 + x}{10 + x} = 0.6\n]", "This is a rational equation where a simple rational expression equals a decimal. Solving it involves isolating the variable (x), eliminating the denominator, and simplifying algebraically.", "---", "## Step-by-Step Solution", "### Step 1: Eliminate the Fraction", "To remove the fraction, multiply both sides of the equation by the denominator, (10 + x):", "[\n(10 + x) \cdot \frac{4 + x}{10 + x} = 0.6 \cdot (10 + x)\n]", "Since (10 + x <br/>\neq 0) (which means (x <br/>\neq -10)), we can cancel the denominator:", "[\n4 + x = 0.6(10 + x)\n]", "---", "### Step 2: Expand the Right Side", "Distribute the 0.6 across the parentheses:", "[\n4 + x = 6 + 0.6x\n]", "---", "### Step 3: Collect Like Terms", "Move all terms containing (x) to one side and constant terms to the other:", "[\n4 + x - 0.6x = 6\n]", "Simplify:", "[\n4 + 0.4x = 6\n]", "Subtract 4 from both sides:", "[\n0.4x = 2\n]", "---", "### Step 4: Solve for (x)", "Divide both sides by 0.4 (or equivalently, multiply by ( \frac{10}{4} = 2.5 )):", "[\nx = \frac{2}{0.4} = 5\n]", "---", "### Step 5: Check the Solution", "It’s crucial to verify the solution by substituting (x = 5) back into the original equation:", "[\n\frac{4 + 5}{10 + 5} = \frac{9}{15} = 0.6\n]", "The left-hand side equals the right-hand side, confirming the solution is correct.", "---", "## Why This Equation Matters", "Equations like ( \frac{4 + x}{10 + x} = 0.6 ) appear frequently in real-life situations:", "- Finance: Calculating break-even points where ratios represent profit rates over costs.\n- Physics: Modeling resistance or efficiency in electrical circuits.\n- Statistics: Simplifying proportions in probability and sampling.", "By solving such equations, you build foundational skills for tackling more complex models used in science, engineering, and economics.", "---", "## Key Concepts Recap", "- Rational Equations: Equations containing fractions with variables in the denominator.\n- Eliminating Denominators: Multiply both sides by the common denominator to simplify.\n- Isolating the Variable: Move all (x)-terms and constants to opposite sides.\n- Checking Solutions: Important to prevent extraneous answers and confirm correctness.\n- Working with Decimals: Converting 0.6 to a fraction (( \frac{3}{5} )) can simplify calculations, especially when exact decimals are used.", "---", "## Conclusion", "Solving ( \frac{4 + x}{10 + x} = 0.6 ) demonstrates core algebraic techniques essential for mastering equations involving fractions and decimals. Practice bouncing back and forth between steps—clear reasoning leads to accurate results. If you’re preparing for exams, tackling problems like this builds confidence and precision in algebraic manipulation. Stay consistent with practice, and soon these equations will feel intuitive.", "---", "Keywords for SEO:\n( \frac{4 + x}{10 + x} = 0.6 ) equation, solve rational equations, algebra tutoring, solving equations with decimals, linear equation example, equation solving strategy, step-by-step algebra, real-world math applications, check solution validity.", "---", "Unlock the power of equations today—start solving like a pro!"]