Check: LHS = |4/3 - 5| = |-11/3| = 11/3, RHS = |2/3 + 3| = |11/3| = 11/3 → valid.
["Understanding Absolute Value Equality: Proving LHS = RHS with |4/3 − 5| = |2/3 + 3|", "In mathematics, verifying logical equality through absolute values helps strengthen understanding of expressions involving magnitude and distance from zero. Consider the equation:", "[\n| \ frac{4}{3} - 5 | = | \ frac{2}{3} + 3 |\n]", "At first glance, evaluating both sides might seem straightforward—but let’s break it down step by step to confirm the validity of this equality using absolute value properties.", "---", "### Step 1: Simplify the expression inside the LHS", "We begin with the left-hand side:", "[\n|\ frac{4}{3} - 5|\n]", "Write 5 as a fraction with denominator 3:", "[\n5 = \ frac{15}{3}\n]", "Now subtract:", "[\n\ frac{4}{3} - \ frac{15}{3} = \ frac{4 - 15}{3} = \ frac{-11}{3}\n]", "Take the absolute value:", "[\n|\ frac{-11}{3}| = \ frac{11}{3}\n]", "So,\n[\nLHS = \left| \frac{4}{3} - 5 \right| = \frac{11}{3}\n]", "---", "### Step 2: Simplify the expression inside the RHS", "Now examine the right-hand side:", "[\n|\ frac{2}{3} + 3|\n]", "Write 3 as (\ frac{9}{3}):", "[\n\ frac{2}{3} + \ frac{9}{3} = \ frac{11}{3}\n]", "Take absolute value (positive since the result is positive):", "[\n|\ frac{11}{3}| = \frac{11}{3}\n]", "So,\n[\nRHS = |\ frac{2}{3} + 3| = \frac{11}{3}\n]", "---", "### Step 3: Compare both sides", "We now have:", "[\n\ ext{LHS} = \frac{11}{3}, \quad \ ext{RHS} = \frac{11}{3}\n]", "Thus,", "[\n| \ frac{4}{3} - 5 | = | \ frac{2}{3} + 3 |\n]", "is valid.", "---", "### Why This Matters: The Power of Absolute Value Equality", "Absolute values capture magnitude regardless of sign, which makes equality testing reliable in many branches of math—whether in algebra, inequalities, geometry, or error analysis. This example illustrates how careful simplification and proper handling of fractions ensure accurate verification.", "Key takeaways:\n- Convert mixed numbers to fractions for consistent computation.\n- Apply absolute value rules carefully—inside or outside after simplification.\n- Always check both sides independently in verify tests.", "Memorizing or proving such identities strengthens numerical reasoning and confidence in solving equations involving absolute values.", "---", "Try this with other expressions—use common denominators, simplify first, then apply absolute value rules. Verifying equality step-by-step makes math clearer and prevents errors!"]
