Add 5: \( -2 \leq 2x \leq 12 \).

["# Understanding the Inequality: Solve ( -2 \leq 2x \leq 12 )", "Solving inequalities is a fundamental skill in algebra, especially when dealing with real-world problems involving ranges and constraints. One commonly encountered inequality pattern is the compound inequality:\n[ -2 \leq 2x \leq 12 ]", "This inequality sets bounds on a variable ( x ), requiring you to find all values of ( x ) that satisfy both conditions. Whether you're analyzing data trends, optimizing resources, or solving physics problems, mastering how to solve compound inequalities is essential. In this article, we’ll walk through solving ( -2 \leq 2x \leq 12 ) step-by-step and provide practical applications to reinforce your understanding.", "---", "## What Does the Inequality Mean?", "The expression\n[ -2 \leq 2x \leq 12 ]\nis a compound inequality stating that ( 2x ) lies between (-2) and (12), inclusive. This means that ( 2x ) can be any value from (-2) up to and including 12. To solve for ( x ), we’ll isolate ( x ) by dividing all parts of the inequality by 2—keeping in mind that dividing by a positive number preserves the inequality directions.", "---", "## Step-by-Step Solution", "### Step 1: Divide all parts by 2\n[\n\frac{-2}{2} \leq \frac{2x}{2} \leq \frac{12}{2}\n]", "### Step 2: Simplify\n[\n-1 \leq x \leq 6\n]", "### Final Result\nThe solution in interval notation is:\n[ [-1, 6] ]\nand in set-builder notation:\n[ { x \mid -1 \leq x \leq 6 } ]", "This means ( x ) ranges from (-1) to (6), including both endpoints.", "---", "## Verifying the Solution", "To ensure correctness, test values within and outside the range:", "- Test ( x = -1 ):\n ( 2x = -2 ). Does (-2 \leq -2 \leq 12) hold? Yes.", "- Test ( x = 0 ):\n ( 2x = 0 ). Is ( -2 \leq 0 \leq 12 )? Yes.", "- Test ( x = 6 ):\n ( 2x = 12 ). Does (-2 \leq 12 \leq 12)? Yes.", "- Test ( x = -2 ):\n ( 2x = -4 ). Is (-2 \leq -4 \leq 12)? No (fails the left inequality).", "- Test ( x = 7 ):\n ( 2x = 14 ). Is (-2 \leq 14 \leq 12)? No (fails the right inequality).", "Only values between (-1) and (6) satisfy the original inequality.", "---", "## Key Takeaways", "- Compound inequalities like ( -2 \leq 2x \leq 12 ) can be solved by breaking them into two separate inequalities.\n- Dividing by a positive constant preserves inequality directions; dividing by a negative reverses them (not applicable here).\n- The solution ( -1 \leq x \leq 6 ) represents a closed interval, meaning endpoints are included.\n- Always verify endpoints and values outside the interval to confirm correctness.", "---", "## Real-World Applications", "Understanding how to solve ( -2 \leq 2x \leq 12 ) translates to solving practical problems, such as:", "- Temperature ranges: If a process operates safely between (-2^\circ F) and (12^\circ F) when scaled by 2, find the original temperature range.\n- Finance: Determine income or budget limits constrained between fixed values.\n- Physics: Model motion where displacement lies between (-2) and (12) units when scaled by a factor of 2.", "---", "## Conclusion", "Solving the inequality ( -2 \leq 2x \leq 12 ) yields ( -1 \leq x \leq 6 ), a clear interval representing all valid values of ( x ). Mastering such inequalities builds a strong foundation for algebraic reasoning and real-world problem-solving. Practice breaking compound inequalities into manageable steps, verify solutions carefully, and explore diverse applications to solidify your understanding.", "For further study, review how to graph inequalities on a number line and tackle other compound forms, including when integers or inequalities are part of the expressions.", "---", "Keywords: solve ( -2 \leq 2x \leq 12 ), compound inequality, algebra quiz, solving inequalities, real-world applications, interval notation, step-by-step algebra."]