\[ = rac{1}{2} \left| 3 + 4 - 20
["Understanding the Expression: How to Simplify $ \frac{1}{2} \left| 3 + 4 - 20 \right|", "When solving mathematical expressions involving absolute values and fractions, clarity is key. One common problem students encounter is evaluating expressions like:", "[\n= \frac{1}{2} \left| 3 + 4 - 20 \right|\n]", "This article explains step-by-step how to break down and solve this expression, with a focus on improving your understanding of absolute value and fraction operations in algebra.", "---", "### Step 1: Simplify Inside the Absolute Value", "The expression begins with the sum inside the absolute value:", "[\n3 + 4 - 20\n]", "First, perform the addition:", "[\n3 + 4 = 7\n]", "Now subtract:", "[\n7 - 20 = -13\n]", "So the expression becomes:", "[\n\left| -13 \right|\n]", "The absolute value of a negative number is its positive counterpart:", "[\n\left| -13 \right| = 13\n]", "---", "### Step 2: Apply the Fraction", "Now multiply the absolute value by $\frac{1}{2}$:", "[\n\frac{1}{2} \ imes 13 = \frac{13}{2}\n]", "---", "### Final Answer", "[\n\boxed{\frac{13}{2}}\n]", "---", "### Why This Matters: Absolute Value and Fractions in Math", "Absolute value signs ($\left| x \right|$) ensure non-negative results, making them useful in distances, errors, and inequalities. When combined with fractions—like in this expression—mastery of order of operations and absolute value interpretation prevents common mistakes.", "Knowing how to simplify expressions of this form helps students confidently approach similar problems in algebra, calculus, and applied mathematics, especially in fields requiring precise calculations such as physics, engineering, and statistics.", "---", "Summary:\nEvaluating $\frac{1}{2} \left| 3 + 4 - 20 \right|$ simplifies step-by-step: inside the absolute value, compute the sum to get $-13$, then apply the absolute value to get $13$, and finally divide by $2$ to reach $\frac{13}{2}$. Understanding each step enables clearer problem-solving and stronger mathematical foundations."]
