Largest: solve \( 7k + 3 \leq 150 \Rightarrow 7k \leq 147 \Rightarrow k \leq 21 \)

["# Solving the Inequality: ( 7k + 3 \leq 150 ) — Step-by-Step Guide to Find the Largest Integer Solution", "Maths inequalities are essential tools in algebra, helping students and problem solvers understand boundaries and constraints. One common problem is solving linear inequalities like ( 7k + 3 \leq 150 ) to find the largest possible integer value of ( k ). In this article, we break down the step-by-step solution, explain the logic, and highlight practical applications.", "---", "## Understanding the Inequality: ( 7k + 3 \leq 150 )", "The inequality ( 7k + 3 \leq 150 ) states that the expression ( 7k + 3 ) is less than or equal to 150. Our goal is to isolate ( k ) and determine its maximum integer value.", "---", "## Step 1: Subtract 3 from Both Sides", "To eliminate the constant term, subtract 3 from both sides:", "[\n7k + 3 - 3 \leq 150 - 3\n\Rightarrow 7k \leq 147\n]", "This simplifies the problem to a solvable form.", "---", "## Step 2: Divide Both Sides by 7", "Next, divide both sides by 7 to isolate ( k ):", "[\n\frac{7k}{7} \leq \frac{147}{7}\n\Rightarrow k \leq 21\n]", "This shows that ( k ) must be less than or equal to 21.", "---", "## Why Is ( k \leq 21 ) the Largest Integer Solution?", "The inequality ( k \leq 21 ) means ( k ) can take any real number less than or equal to 21. However, since we are often asked for the largest integer solution in practical applications (e.g., testing discrete values, resource allocation), the answer is 21 because:\n- 21 satisfies the inequality: ( 7(21) + 3 = 147 + 3 = 150 \leq 150 ).\n- The next integer, 22, gives ( 7(22) + 3 = 154 + 3 = 157 > 150 ), which violates the condition.", "---", "## Final Answer: The Largest Integer Solution", "[\n\boxed{21}\n]", "Thus, the largest integer value of ( k ) satisfying ( 7k + 3 \leq 150 ) is 21.", "---", "## Practical Applications of Solving Linear Inequalities", "Linear inequalities like this are widely used in:\n- Budgeting: Limiting spending within a maximum allowed amount.\n- Science: Defining safe concentration or dosage ranges.\n- Engineering: Setting constraints in design and analysis.", "Understanding step-by-step solving techniques helps build a strong foundation for more advanced algebra and real-world modeling.", "---", "## Bonus: Quick Check of the Solution", "Let’s verify:\n- For ( k = 21 ): ( 7(21) + 3 = 147 + 3 = 150 \leq 150 ) ✅\n- For ( k = 20 ): ( 7(20) + 3 = 140 + 3 = 143 \leq 150 ) ✅ (but smaller)\nThis confirms that 21 is indeed the maximum integer.", "---", "## Summary", "Solving ( 7k + 3 \leq 150 ) involves:\n1. Subtracting 3 from both sides\n2. Dividing both sides by 7\n3. Concluding that ( k \leq 21 )", "The largest integer solution is ( k = 21 ), a valuable result in mathematical reasoning and problem solving.", "Start mastering inequalities today — your future math success depends on it! 📚✅"]