Find the number of integer solutions to the inequality \( |2x - 5| \leq 7 \).
["# Finding the Number of Integer Solutions to the Inequality ( |2x - 5| \leq 7 )", "The inequality ( |2x - 5| \leq 7 ) is a common type of absolute value inequality used to determine key integer solutions on the number line. Understanding how to solve such inequalities not only strengthens algebraic skills but also helps in real-world problem modeling. This article explains step-by-step how to find all integer values of ( x ) that satisfy this condition, and provides insight into counting those solutions efficiently.", "---", "## What Does ( |2x - 5| \leq 7 ) Mean?", "The absolute value inequality ( |A| \leq B ), where ( B \geq 0 ), means that ( A ) lies between ( -B ) and ( B ). Applying this to our expression:", "[\n|2x - 5| \leq 7\n]", "means:", "[\n-7 \leq 2x - 5 \leq 7\n]", "This compound inequality lets us solve for ( x ) by isolating the variable.", "---", "## Step-by-Step Solution", "### Step 1: Remove the Absolute Value", "Starting with:", "[\n-7 \leq 2x - 5 \leq 7\n]", "Add 5 to all three parts:", "[\n-7 + 5 \leq 2x - 5 + 5 \leq 7 + 5\n]", "[\n-2 \leq 2x \leq 12\n]", "### Step 2: Solve for ( x )", "Divide every part by 2:", "[\n\frac{-2}{2} \leq \frac{2x}{2} \leq \frac{12}{2}\n]", "[\n-1 \leq x \leq 6\n]", "So, all real numbers ( x ) satisfying the inequality lie in the closed interval ( [-1, 6] ).", "---", "## Finding Integer Solutions", "Now, we need the integer values of ( x ) between ( -1 ) and ( 6 ), inclusive. The integers satisfying this are:", "[\n-1, 0, 1, 2, 3, 4, 5, 6\n]", "Count them:\nFrom ( -1 ) to ( 6 ), inclusive, there are:", "[\n6 - (-1) + 1 = 8 \ ext{ integer solutions}\n]", "---", "## Visualizing the Solution", "Plotting the number line:", "- Inequality holds for ( x \in [-1, 6] ), meaning from ( -1 ) up to and including ( 6 ).\n- All integers within this range are valid solutions.", "---", "## Why Count Integer Solutions?", "Integers are crucial in discrete mathematics, computer science, and real-life applications such as counting discrete objects, scheduling, or resource allocation. Knowing how many integer solutions exist makes optimization and planning easier.", "---", "## Final Answer", "The inequality ( |2x - 5| \leq 7 ) has 8 integer solutions, namely:", "[\nx = -1, 0, 1, 2, 3, 4, 5, 6\n]", "---", "## Summary", "- Solve absolute value inequalities by converting to compound inequalities.\n- Isolate ( x ) by adding/subtracting constants and dividing.\n- Identify the closed interval of real solutions ( [-1, 6] ).\n- Count integers within this interval to get the number of solutions.", "Mastering such techniques improves mathematical fluency and problem-solving efficiency.", "---", "Keywords: integer solutions, absolute value inequality, ( |2x - 5| \leq 7 ), solving inequalities, number theory, counting solutions, algebra, discrete mathematics.\nHeader tags optimized for SEO: Finding integer solutions to |2x−5|≤7, step-by-step guide to |2x−5|≤7, how many integers satisfy |2x−5|≤7, solving absolute value inequalities."]
