Find the sum of all integer solutions to the inequality |2x − 5| < 9.
["Finding the Sum of All Integer Solutions to the Inequality |2x – 5| < 9", "When solving inequalities involving absolute values, one common challenge is identifying all possible integer solutions and then computing their sum. In this article, we explore how to find and sum all integer solutions to the inequality:", "[\n|2x - 5| < 9\n]", "---", "### Understanding the Absolute Value Inequality", "The absolute value inequality ( |A| < B ), where ( B > 0 ), is equivalent to:", "[\n-B < A < B\n]", "Applying this rule to ( |2x - 5| < 9 ), we rewrite it as:", "[\n-9 < 2x - 5 < 9\n]", "Now solve this compound inequality step by step.", "---", "### Step 1: Solve the Compound Inequality", "Start with:", "[\n-9 < 2x - 5 < 9\n]", "Add 5 to all parts:", "[\n-9 + 5 < 2x < 9 + 5\n]\n[\n-4 < 2x < 14\n]", "Now divide all parts by 2:", "[\n-2 < x < 7\n]", "This tells us that ( x ) lies strictly between (-2) and (7), meaning ( x ) is greater than (-2) and less than (7).", "---", "### Step 2: Identify All Integer Solutions", "We seek all integers satisfying ( -2 < x < 7 ).", "The integers in this range are:", "[\n-1, 0, 1, 2, 3, 4, 5, 6\n]", "Note: ( x ) cannot be (-2) (not greater than (-2)) or (7) (not less than (7)).", "---", "### Step 3: Sum All Integer Solutions", "Now compute the sum:", "[\n(-1) + 0 + 1 + 2 + 3 + 4 + 5 + 6\n]", "Group terms for clarity:", "[\n(-1 + 1) + (0) + (2 + 6) + (3 + 5) + 4 = 0 + 0 + 8 + 8 + 4 = 20\n]", "Alternatively, simply add step by step:", "[\n-1 + 0 = -1 \\n-1 + 1 = 0 \\n0 + 2 = 2 \\n2 + 3 = 5 \\n5 + 4 = 9 \\n9 + 5 = 14 \\n14 + 6 = 20\n]", "Thus, the total sum is ( 20 ).", "---", "### Summary", "The inequality ( |2x - 5| < 9 ) holds for integers ( x = -1, 0, 1, 2, 3, 4, 5, 6 ). Adding these values gives a total sum of:", "[\n\boxed{20}\n]", "---", "### Why This Matters for Problem Solvers", "Absolute value inequalities often define bounded intervals on the number line. Identifying integer solutions and computing their sum is useful inoptimization, modeling real-world constraints, or educational exercises in algebra. Mastering techniques like converting ( |A| < B ) to ( -B < A < B ) and extracting integer bounds strengthens algebraic reasoning and inequality-solving skills.", "---", "Keywords: integer solutions, absolute value inequality, find sum of integers, |2x – 5| < 9, algebra tutorial, solving inequalities, sum of integers."]
