This inequality becomes \( -7 \leq 2x - 5 \leq 7 \).

["Understanding the Inequality: ( -7 \leq 2x - 5 \leq 7 ) – Solving & Explaining Step-by-Step", "Inequalities often arise in real-world applications, from modeling budgets to optimizing resources. One commonly encountered type is compound inequalities like ( -7 \leq 2x - 5 \leq 7 ). Solving this not only sharpens algebraic skills but also strengthens logical reasoning. In this article, we’ll break down how to solve ( -7 \leq 2x - 5 \leq 7 ) step-by-step, interpret its meaning, and explore practical applications.", "---", "### What Is the Inequality ( -7 \leq 2x - 5 \leq 7 )?", "This is a compound inequality expressing two conditions:", "- ( 2x - 5 \geq -7 )\n- ( 2x - 5 \leq 7 )", "In words, the expression ( 2x - 5 ) lies between –7 and 7, inclusive. Solving this gives all real values of ( x ) for which this condition holds.", "---", "### Step-by-Step Solution", "Start with the compound inequality:", "[\n-7 \leq 2x - 5 \leq 7\n]", "We’ll solve each part individually and then combine the solution.", "Step 1: Add 5 to all three parts", "Adding 5 neutralizes the –5 on the left and right:", "[\n-7 + 5 \leq 2x - 5 + 5 \leq 7 + 5\n]", "[\n-2 \leq 2x \leq 12\n]", "Step 2: Divide all parts by 2", "To isolate ( x ), divide by 2:", "[\n\frac{-2}{2} \leq \frac{2x}{2} \leq \frac{12}{2}\n]", "[\n-1 \leq x \leq 6\n]", "---", "### Final Solution", "The solution set is all real numbers ( x ) such that:", "[\n\boxed{ -1 \leq x \leq 6 }\n]", "---", "### Geometric Interpretation: Number Line Representation", "Graphically, this means ( x ) lies in the closed interval ([-1, 6]) on the number line, including endpoints -1 and 6. This visual aid helps interpret inequality ranges intuitively—any ( x ) between –1 and 6 satisfies the original compound inequality.", "---", "### Applications of This Inequality", "Understanding inequalities like ( -7 \leq 2x - 5 \leq 7 ) is crucial in multiple domains:", "- Financial Planning: Determining acceptable ranges for budgets or price limits\n Example: If a product’s cost after a 10% fee is modeled by ( 2x - 5 ), and must stay between –7 and 7, solve for ( x ) to ensure feasibility.", "- Engineering & Quality Control: Maintaining component dimensions within tolerance levels\n If measured variation follows ( |2x - 5| \leq 7 ), solving gives usable variable values.", "- Health and Safety: Monitoring environmental conditions, such as temperature tolerance for sensitive equipment defined by ( -7 \leq T \leq 7 ).", "---", "### Why Solving Compound Inequalities Matters", "Mastering problems like ( -7 \leq 2x - 5 \leq 7 ) builds:", "- Algebraic fluency in manipulating inequalities respecting directionality\n- Logical analysis tools applicable beyond math—into programming, design, and decision-making\n- Confidence in interpreting real-life constraints expressed numerically", "---", "### Summary", "To solve ( -7 \leq 2x - 5 \leq 7 ):", "1. Rewrite as two separate inequalities: ( 2x - 5 \geq -7 ) and ( 2x - 5 \leq 7 )\n2. Add 5 to both sides: ( -2 \leq 2x \leq 12 )\n3. Divide by 2: ( -1 \leq x \leq 6 )\n4. Express solution as a closed interval on the number line", "This approach not only provides exact values but also deepens understanding of inequality behavior and its broad applicability.", "---", "### Further Reading", "- Always remember: dividing by a negative reverses inequality signs (important caution point)\n- Practice with real-world contexts to strengthen application skills\n- Use graphing tools and number lines to visualize solution sets", "---", "Tagline: Perfecting your math toolkit—one inequality at a time! Understand ( -7 \leq 2x - 5 \leq 7 ), unlock practical solutions, and apply algebraic logic confidently."]