Divide by 2: \( -1 \leq x \leq 6 \).

["# Understanding and Solving the Inequality: Divide by 2 with Bounds ( -1 \leq x \leq 6 )", "When solving inequalities, especially those involving division, interpreting the bounds correctly is crucial. One common problem involves dividing variables within a given interval, such as ( -1 \leq x \leq 6 ), bounded by division by 2. This article breaks down how to properly divide this expression, explains the solution process, and clarifies the meaning of the final result.", "## What Does ( -1 \leq x \leq 6 ) Mean?", "The inequality ( -1 \leq x \leq 6 ) defines a closed interval from (-1) to (6), including both endpoints. This means ( x ) can be any real number between (-1) and (6), including (-1) and (6) themselves.", "## Step 1: Dividing the Inequality by 2", "To solve after division, we must divide every part of the inequality by 2. Since 2 is a positive number, the direction of the inequality remains unchanged — a key rule in algebra: multiplying or dividing both sides of an inequality by a positive number does not reverse the inequality sign.", "[\n\frac{-1}{2} \leq \frac{x}{2} \leq \frac{6}{2}\n]", "Simplify each term:", "[\n-0.5 \leq \frac{x}{2} \leq 3\n]", "This simplifies further to:", "[\n-0.5 \leq \frac{x}{2} \leq 3\n]", "## Step 2: Solving for ( x )", "To isolate ( x ), multiply every term by 2:", "[\n2(-0.5) \leq 2\left(\frac{x}{2}\right) \leq 2(3)\n]", "[\n-1 \leq x \leq 6\n]", "Interestentially, dividing by 2 reversed neither the bounds nor maintained the original inequality direction. This means the solution interval remains exactly the same — it’s just expressed in a different form.", "## Why Does the Interval Stay Fixed?", "Because division by a positive number preserves the original inequalities. The values ( -1 ) and ( 6 ) stay valid bounds, semi-scaled by 2. This highlights an essential algebraic principle: dividing an inequality by a positive scalar does not flip inequality signs — only multiplying does.", "## Visualizing the Solution", "Plotting (-1 \leq x \leq 6) on a number line, with points at (-1) and (6) marked and shaded, confirms all values in between (including endpoints) satisfy the original inequality. After dividing by 2, this interval maps directly without change.", "## Key Takeaways", "- Always maintain inequality direction when dividing by a positive number.\n- The bounds (-1) and (6) stay unchanged, but the expression becomes ( \frac{x}{2} ) inside the interval.\n- Understanding the original domain is essential before and after scaling.", "## Practical Tips for Applying This Concept", "When solving real-world problems involving constraints (e.g., temperature ranges, distances, scalings), recognizing intervals and careful division prevents errors. Always:\n1. Write out the inequality clearly.\n2. Ensure consistent sign usage.\n3. Verify dimensional consistency (e.g., dividing length by 2 keeps units in original units).\n4. Recheck the interval bounds post-operation.", "## Conclusion", "The expression ( -1 \leq x \leq 6 ), under division by 2 (a positive value), transforms into ( -0.5 \leq \frac{x}{2} \leq 3 ), which simplifies back to the original bounds. This consistency reinforces proper algebraic handling and confirms that dividing a closed interval by a positive scalar retains its integrity. Mastering this ensures accurate solutions in mathematics and applied fields alike.", "---", "Keywords: Divide by 2, inequality solution, ( -1 \leq x \leq 6 ), algebraic manipulation, understanding bounds, real numbers interval, solving linear inequalities.\nMeta description: Understand how dividing the inequality ( -1 \leq x \leq 6 ) by 2 preserves the interval’s bounds and direction. Learn step-by-step solving with best practices in algebra."]