The sum of all real solutions to the equation \( |2x - 5| = |x + 3| \) is?
["The Sum of All Real Solutions to the Equation |2x – 5| = |x + 3|: A Clear Breakdown", "---", "### Understanding Absolute Value Equations", "Absolute value equations involve expressions with modulus signs, meaning the distance from zero on the number line. When solving equations like ( |A| = |B| ), a fundamental rule applies:\n[ |A| = |B| \implies A = B \quad \ ext{or} \quad A = -B ]", "This means we split the equation into two cases to find all real solutions.", "---", "### Solving ( |2x - 5| = |x + 3| )", "Start with the equation:", "[\n|2x - 5| = |x + 3|\n]", "Using the rule above, we consider two cases:", "---", "#### Case 1:\n[\n2x - 5 = x + 3\n]", "Subtract ( x ) from both sides:", "[\nx - 5 = 3\n]", "Add 5 to both sides:", "[\nx = 8\n]", "---", "#### Case 2:\n[\n2x - 5 = -(x + 3)\n]", "Distribute the negative sign:", "[\n2x - 5 = -x - 3\n]", "Add ( x ) to both sides:", "[\n3x - 5 = -3\n]", "Add 5 to both sides:", "[\n3x = 2\n]", "Divide by 3:", "[\nx = \frac{2}{3}\n]", "---", "### Verifying Solutions", "It’s essential to check both solutions in the original equation.", "- For ( x = 8 ):\n $ |2(8) - 5| = |16 - 5| = |11| = 11 $\n $ |8 + 3| = |11| = 11 $ ✅", "- For ( x = \frac{2}{3} ):\n $ |2(\frac{2}{3}) - 5| = \left|\frac{4}{3} - \frac{15}{3}\right| = \left|-\frac{11}{3}\right| = \frac{11}{3} $\n $ |\frac{2}{3} + 3| = \left|\frac{2}{3} + \frac{9}{3}\right| = \left|\frac{11}{3}\right| = \frac{11}{3} $ ✅", "Both solutions are correct.", "---", "### Finding the Sum of Real Solutions", "Now, add the two real solutions:", "[\n8 + \frac{2}{3} = \frac{24}{3} + \frac{2}{3} = \frac{26}{3}\n]", "---", "### Conclusion: Final Answer", "The sum of all real solutions to the equation ( |2x - 5| = |x + 3| ) is:", "[\n\boxed{\frac{26}{3}}\n]", "---", "### Bonus: Insight", "Geometrically, solving ( |2x - 5| = |x + 3| ) means finding points where two distances from zero are equal — this happens at two points on the number line, symmetric in their distances. The sum of these points simplifies neatly to ( \frac{26}{3} ), confirming algebraic precision.", "---", "Keywords:\nsum of real solutions, |2x – 5| = |x + 3|, absolute value equation, solving absolute equations, step-by-step solution, real solutions, math tutorial, equation sum, algebra tips"]
