Set up the equation: \( \frac{3 + x}{10 + x} = 0.5 \).

["Set Up and Solve the Equation: ( \frac{3 + x}{10 + x} = 0.5 )", "Understanding how to set up and solve rational equations is essential for success in algebra. One common challenge is solving equations involving fractions with variables in both the numerator and denominator. In this article, we’ll walk through how to properly set up the equation ( \frac{3 + x}{10 + x} = 0.5 ), explaining each step clearly for better comprehension and problem-solving skills.", "---", "### Why Set Up This Equation?", "The equation ( \frac{3 + x}{10 + x} = 0.5 ) appears frequently in science, economics, and everyday problem-solving. It models proportional relationships and can represent rates, percentages, or ratios. Mastering its setup reinforces foundational algebra skills necessary for higher-level math.", "---", "### Step-by-Step Setup", "Step 1: Identify the Fractional Expression", "The left-hand side, ( \frac{3 + x}{10 + x} ), is a rational expression — a fraction with polynomials in the numerator and denominator.", "Step 2: Set the Expression Equal to Given Value", "We know this expression equals ( 0.5 ), so we write:", "[\n\frac{3 + x}{10 + x} = 0.5\n]", "This setup assumes ( 10 + x <br/>\neq 0 ), because division by zero is undefined.", "---", "### Solve the Equation", "Now we solve for ( x ):", "Step 1: Eliminate the Denominator", "Multiply both sides of the equation by ( 10 + x ):", "[\n3 + x = 0.5(10 + x)\n]", "Step 2: Distribute the Right Side", "[\n3 + x = 5 + 0.5x\n]", "Step 3: Move All ( x )-Terms to One Side and Constants to the Other", "Subtract ( 0.5x ) from both sides:", "[\n3 + 0.5x = 5\n]", "Subtract 3 from both sides:", "[\n0.5x = 2\n]", "Step 4: Solve for ( x )", "Divide both sides by 0.5:", "[\nx = \frac{2}{0.5} = 4\n]", "---", "### Final Step: Check the Solution", "Never skip verification! Plug ( x = 4 ) back into the original equation:", "[\n\frac{3 + 4}{10 + 4} = \frac{7}{14} = 0.5\n]", "The left side equals the right side, confirming the solution is correct.", "---", "### Conclusion: Key Takeaways", "Setting up the equation ( \frac{3 + x}{10 + x} = 0.5 ) correctly is the first crucial step toward solving rational equations. By:", "- Expressing the unknown relationship as a fraction,\n- Isolating the fraction on one side, and\n- Eliminating the denominator properly,", "we arrived at ( x = 4 ). Always remember to eliminate restrictions like ( 10 + x <br/>\neq 0 ), which in this case means ( x <br/>\neq -10 )—a practical constraint to keep in mind.", "Master this method, and you’ll confidently tackle similar equations in math, science, and real-world applications.", "---", "Keywords: set up equation, solve rational equation, ( \frac{3 + x}{10 + x} = 0.5 ), algebra, how to solve fractions, check solution, linear equation, step-by-step math", "Meta Description: Learn how to set up and solve the equation ( \frac{3 + x}{10 + x} = 0.5 ), including algebraic steps, verification, and key problem-solving tips. Perfect for algebra students and teachers."]