#### 96Question: What is the sum of all values of $ x $ for which $ \sqrt{(x - 3)^2} = 2x - 5 $?
["Understanding $ \sqrt{(x - 3)^2} = 2x - 5 $: A Step-by-Step Breakdown and Sum of All Valid Solutions", "When solving equations involving square roots and absolute values, clarity and precision are essential. One commonly encountered problem is:", "$ \sqrt{(x - 3)^2} = 2x - 5 $", "But what does this equation really mean, and what are the valid solutions? More specifically, if multiple values of $ x $ satisfy it, what is their sum?", "---", "### What Does $ \sqrt{(x - 3)^2} $ Mean?", "The square root of a squared expression, $ \sqrt{(x - 3)^2} $, simplifies to the absolute value:", "$$\n\sqrt{(x - 3)^2} = |x - 3|\n$$", "So the equation becomes:", "$$\n|x - 3| = 2x - 5\n$$", "Absolute values introduce conditional behavior — the equation splits into cases based on the sign of $ x - 3 $.", "---", "### Step 1: Consider Cases Based on the Expression Inside the Absolute Value", "#### Case 1: $ x - 3 \geq 0 $ ⟹ $ x \geq 3 $\nIn this case, $ |x - 3| = x - 3 $, so:", "$$\nx - 3 = 2x - 5\n$$", "Solve for $ x $:", "$$\nx - 3 = 2x - 5 \Rightarrow -3 + 5 = 2x - x \Rightarrow x = 2\n$$", "But wait — this solution $ x = 2 $ does not satisfy $ x \geq 3 $. So no valid solution from this case.", "---", "#### Case 2: $ x - 3 < 0 $ ⟹ $ x < 3 $\nHere, $ |x - 3| = -(x - 3) = 3 - x $, so:", "$$\n3 - x = 2x - 5\n$$", "Solve:", "$$\n3 + 5 = 2x + x \Rightarrow 8 = 3x \Rightarrow x = \frac{8}{3}\n$$", "Check if this value satisfies $ x < 3 $:\nSince $ \frac{8}{3} \approx 2.67 $, it does satisfy $ x < 3 $. Also, verify the original right-hand side:", "$$\n2x - 5 = 2\left(\frac{8}{3}\right) - 5 = \frac{16}{3} - \frac{15}{3} = \frac{1}{3} > 0\n$$", "Since the square root function always returns a non-negative value, $ 2x - 5 $ must be $ \geq 0 $. $ \frac{1}{3} \geq 0 $, so condition holds.", "Thus, $ x = \frac{8}{3} $ is a valid solution.", "---", "### Step 2: Are There Any Other Solutions?", "We’ve considered all real cases: only $ x = \frac{8}{3} $ satisfies the equation under valid domain conditions. No other values work.", "---", "### Step 3: Sum of All Valid Solutions", "Since only one valid solution exists — $ x = \frac{8}{3} $ — the sum of all such values is simply $ \frac{8}{3} $.", "---", "### Final Answer", "$$\n\boxed{\frac{8}{3}}\n$$", "This problem highlights how absolute values transform square roots into piecewise equations and emphasizes the importance of checking domain restrictions and non-negativity in solutions.", "---", "Keywords for SEO: \n$ \sqrt{(x - 3)^2} $ equation, solve $ \sqrt{(x - 3)^2} = 2x - 5 $, sum of solutions, absolute value equation, case analysis, valid $ x $ values, step-by-step algebra, mathematics tutoring guide", "Meta Description:\nExplore how to solve $ \sqrt{(x - 3)^2} = 2x - 5 $, analyze cases using absolute value, and find the sum of all valid solutions. Learn key math techniques with clear, step-by-step reasoning.", "Main Keywords:\n$ \sqrt{(x - 3)^2} = 2x - 5 $, solution, sum of values, absolute value equation, algebra problem, math solution guide"]
