Question: Find the largest value of $ x $ that satisfies $ |2x - 5| + |x + 3| = 12 $.
["# Find the Largest Value of $ x $ That Satisfies $ |2x - 5| + |x + 3| = 12 $", "Understanding absolute value equations can be challenging, but solving $ |2x - 5| + |x + 3| = 12 $ step by step reveals clear intervals to find the solution. This guide explains how to find the largest value of $ x $ that satisfies the equation by analyzing critical points where expressions inside the absolute values change sign.", "## Understanding the Absolute Value Equation", "The expression $ |2x - 5| + |x + 3| = 12 $ involves absolute values, which behave differently depending on whether the inside expression is positive or negative. Absolute values break equations into piecewise segments, so identifying where each expression inside the bars switches sign is essential.", "Define the key expressions:\n- $ 2x - 5 = 0 $ → $ x = \frac{5}{2} = 2.5 $\n- $ x + 3 = 0 $ → $ x = -3 $", "These points divide the number line into three intervals:\n1. $ x < -3 $\n2. $ -3 \leq x < 2.5 $\n3. $ x \geq 2.5 $", "We will solve the equation separately in each interval to eliminate absolute values and find valid solutions.", "## Step 1: Solve in Interval $ x < -3 $", "In this range:\n- $ 2x - 5 < 0 $ → $ |2x - 5| = -(2x - 5) = -2x + 5 $\n- $ x + 3 < 0 $ → $ |x + 3| = -(x + 3) = -x - 3 $", "Substitute into the equation:\n$$\n-2x + 5 + (-x - 3) = 12\n$$\n$$\n-3x + 2 = 12\n$$\n$$\n-3x = 10 \quad \Rightarrow \quad x = -\frac{10}{3} \approx -3.33\n$$", "Check if $ x = -\frac{10}{3} $ satisfies $ x < -3 $:\nSince $ -3.33 < -3 $, this solution is valid.", "## Step 2: Solve in Interval $ -3 \leq x < 2.5 $", "In this range:\n- $ 2x - 5 < 0 $ → $ |2x - 5| = -2x + 5 $\n- $ x + 3 \geq 0 $ → $ |x + 3| = x + 3 $", "Substitute:\n$$\n-2x + 5 + x + 3 = 12\n$$\n$$\n-x + 8 = 12\n$$\n$$\n-x = 4 \quad \Rightarrow \quad x = -4\n$$", "But $ x = -4 $ does not belong to $ -3 \leq x < 2.5 $. So discard this solution.", "## Step 3: Solve in Interval $ x \geq 2.5 $", "In this range:\n- $ 2x - 5 \geq 0 $ → $ |2x - 5| = 2x - 5 $\n- $ x + 3 \geq 0 $ → $ |x + 3| = x + 3 $", "Substitute:\n$$\n2x - 5 + x + 3 = 12\n$$\n$$\n3x - 2 = 12\n$$\n$$\n3x = 14 \quad \Rightarrow \quad x = \frac{14}{3} \approx 4.67\n$$", "Check $ x = \frac{14}{3} \geq 2.5 $:\n$ 2.5 = \frac{5}{2} = 2.5 $, and $ \frac{14}{3} \approx 4.67 > 2.5 $, so valid.", "## Conclusion: Find the Largest Solution", "We found two valid solutions:\n- $ x = -\frac{10}{3} $ from $ x < -3 $\n- $ x = \frac{14}{3} $ from $ x \geq 2.5 $", "The largest value is $ x = \frac{14}{3} $.", "This value satisfies the original equation and belongs to its correct interval, making it the largest solution.", "Final Answer: The largest value of $ x $ satisfying $ |2x - 5| + |x + 3| = 12 $ is $ \boxed{\frac{14}{3}} $.", "---", "### Bonus Insight: Graph Interpretation", "Plotting $ f(x) = |2x - 5| + |x + 3| $ gives a piecewise linear function with minimum at $ x = 2.5 $. The graph increases on both sides, so the equation $ f(x) = 12 $ can have two intersections — one on the left and one on the right — confirming why both intervals yield solutions, with the right one giving the largest $ x $.", "This method balances algebraic rigor with conceptual clarity for solving complex absolute value equations."]
