Solution: The equation $ \sqrt{(x - 3)^2} = 2x - 5 $ simplifies to $ |x - 3| = 2x - 5 $. This gives two cases:

["# Solving the Equation $ \sqrt{(x - 3)^2} = 2x - 5 $: A Step-by-Step Breakdown", "Mistakes happen when solving equations involving absolute values or square roots—especially when simplifying expressions like $ \sqrt{(x - 3)^2} $. One of the most common pitfalls is forgetting to account for the absolute value, which leads to incorrect solutions. In this article, we’ll explore how to properly solve the equation\n$$\n\sqrt{(x - 3)^2} = 2x - 5,\n$$\nshowing exactly how it simplifies to $ |x - 3| = 2x - 5 $, and why splitting into two cases is essential. We’ll walk through the process clearly—so you can confidently solve similar problems without missing critical details.", "---", "## Why Absolute Value Matters in Radical Equations", "The square root of a squared term always yields a non-negative result:\n$$\n\sqrt{(x - 3)^2} = |x - 3|.\n$$\nThis means the left-hand side of the original equation is not just $ x - 3 $, but its absolute value:\n$$\n|x - 3| = 2x - 5.\n$$\nTrigonometric steps or assuming $ x - 3 \geq 0 $ can invalidate the equation for values that make $ 2x - 5 $ negative—genes for extraneous solutions. That’s why recognizing and splitting into cases is crucial.", "---", "## Step 1: Rewrite the Equation with Absolute Value", "Start by rewriting the equation clearly:\n$$\n|x - 3| = 2x - 5.\n$$\nAbsolute values mean the expression inside can be either positive or negative, so we must consider both scenarios.", "---", "## Step 2: Define the Two Key Cases", "The expression $ |x - 3| $ splits into two cases based on the sign of $ x - 3 $:", "### Case 1: $ x - 3 \geq 0 $ → $ x \geq 3 $", "When $ x \geq 3 $, $ |x - 3| = x - 3 $. Substitute into the equation:\n$$\nx - 3 = 2x - 5.\n$$\nNow solve:\n$$\nx - 3 = 2x - 5 \\n-3 + 5 = 2x - x \\n2 = x.\n$$\nSo, $ x = 2 $.", "But wait—we assumed $ x \geq 3 $. Since $ 2 < 3 $, this solution fails the case condition.\nReject $ x = 2 $. No valid solution from Case 1.", "---", "### Case 2: $ x - 3 < 0 $ → $ x < 3 $", "In this region, $ |x - 3| = -(x - 3) = -x + 3 $. So:\n$$\n-x + 3 = 2x - 5.\n$$\nSolve:\n$$\n3 + 5 = 2x + x \\n8 = 3x \\nx = \frac{8}{3}.\n$$", "Check validity: Is $ \frac{8}{3} < 3 $?\nSince $ \frac{8}{3} \approx 2.67 < 3 $, yes. Also verify both sides of the original equation.", "Left: $ \sqrt{\left(\frac{8}{3} - 3\right)^2} = \sqrt{\left(-\frac{1}{3}\right)^2} = \sqrt{\frac{1}{9}} = \frac{1}{3} $.", "Right: $ 2\left(\frac{8}{3}\right) - 5 = \frac{16}{3} - 5 = \frac{16 - 15}{3} = \frac{1}{3} $.", "Both sides match. Valid solution.", "---", "## Final Result", "The only solution to $ \sqrt{(x - 3)^2} = 2x - 5 $ is\n$$\nx = \frac{8}{3}.\n$$", "---", "## Key Takeaways", "- Always rewrite $ \sqrt{a^2} = |a| $ to avoid sign errors.\n- Absolute value requires splitting into cases based on the expression’s sign.\n- Always verify solutions by plugging back into the original equation—especially in weird cases.\n- Ignoring domain constraints (like $ 2x - 5 \geq 0 $) leads to extraneous answers.", "Always remember: absolute value equations may have no solution or one solution, depending on your cases. Never skip checking the case domain!", "---", "### Related Keywords for SEO Optimization", "- Solve $ \sqrt{(x - 3)^2} = 2x - 5\n- How to handle absolute value in equations\n- Step-by-step absolute value equations\n- Why $ |x - 3| = 2x - 5 $ fails without case analysis\n- Absolute value equation case 1 and case 2 explained\n- Common mistakes in solving radical absolute value equations\n- Valid solution check for square root radical equations", "---", "Master absolute values and radicals step by step—here’s how to solve $ |x - 3| = 2x - 5 $ with confidence. No shortcuts, no assumptions—just logic and verification."]