We want $ 1 \leq 7k + 3 \leq 100 $.

["Understanding the Inequality: We Want $ 1 \leq 7k + 3 \leq 100 $", "Solving mathematical inequalities is a fundamental skill that appears in algebra, programming, and real-world problem solving. One common problem type is solving for integer or real values within a bounded range. Today, we explore the inequality:", "$$\n1 \leq 7k + 3 \leq 100\n$$", "This inequality defines a range of values for the variable $ k $, and understanding it helps solve practical problems—from budgeting to algorithm design.", "---", "### Breaking Down the Inequality", "The compound inequality $ 1 \leq 7k + 3 \leq 100 $ can be split into two parts for clearer solving:", "1. $ 1 \leq 7k + 3 $\n2. $ 7k + 3 \leq 100 $", "---", "### Solve the First Inequality:\n$ 1 \leq 7k + 3 $", "Subtract 3 from both sides:", "$$\n1 - 3 \leq 7k \quad \Rightarrow \quad -2 \leq 7k\n$$", "Now divide both sides by 7:", "$$\n-\frac{2}{7} \leq k\n$$", "---", "### Solve the Second Inequality:\n$ 7k + 3 \leq 100 $", "Subtract 3 from both sides:", "$$\n7k \leq 97\n$$", "Divide both sides by 7:", "$$\nk \leq \frac{97}{7} \approx 13.857\n$$", "---", "### Combine Both Results", "From both parts, we get:", "$$\n-\frac{2}{7} \leq k \leq \frac{97}{7}\n$$", "Since $ \frac{97}{7} \approx 13.857 $ and $ -\frac{2}{7} \approx -0.2857 $, the range for $ k $ is:", "$$\n-0.2857 \leq k \leq 13.857\n$$", "---", "### Restricting to Integer Values (Common Context)", "In many applications—such as financial equations or discrete variable problems—$ k $ must be an integer. So we look for integer values of $ k $ such that:", "$$\nk = 0, 1, 2, \dots, 13\n$$", "(We exclude negative integers because $ k \geq -\frac{2}{7} $, and most real-world contexts assume non-negative values.)", "---", "### Check Boundary Values (Optional Verification)", "- For $ k = 0 $: $ 7(0) + 3 = 3 $ ✔️ within $ [1, 100] $\n- For $ k = 13 $: $ 7(13) + 3 = 91 + 3 = 94 $ ✔️\n- For $ k = 14 $: $ 7(14) + 3 = 98 + 3 = 101 $ ❌ exceeds upper bound", "So the largest valid integer is $ k = 13 $.", "---", "### Real-World Applications", "This type of inequality models constraints in:", "- Budgeting: Limiting total costs where each item costs $7 and a base fee of $3 leads to this constraint.\n- Performance limits: Algorithms with scaling factors and base operations often fit within such bounds.\n- Science & Engineering: Sensor readings bounded by physical limits modeled via linear equations.", "---", "### Conclusion", "The inequality $ 1 \leq 7k + 3 \leq 100 $ restricts $ k $ to the integer values between $ 0 $ and $ 13 $, inclusive. Mastering such inequalities strengthens problem-solving skills and enables precise modeling in technology, economics, and science.", "Try solving similar inequalities by breaking them step-by-step—subtraction, division, and domain restriction—especially when working with real-valued variables and constraints.", "---", "Keywords: inequality $ 1 \leq 7k + 3 \leq 100 $, solving linear inequalities, integer values, algebra, real-world applications, inequality breakdown, $ k $ in ranges, math tutorial, problem-solving tips."]