5k - 3 \leq 100 \Rightarrow 5k \leq 103 \Rightarrow k \leq \frac1035 = 20.6 - Decision Point
Understanding and Solving the Inequality: 5k − 3 ≤ 100
Understanding and Solving the Inequality: 5k − 3 ≤ 100
When solving mathematical inequalities, understanding each step ensures clarity and correctness. One common example involves a linear expression that follows a clear sequence of transformations:
5k − 3 ≤ 100 ⇒ 5k ≤ 103 ⇒ k ≤ 20.6
Let’s explore this step-by-step for better comprehension and application in real-world scenarios.
Understanding the Context
Step 1: Start with the Original Inequality
We begin with the inequality:
5k − 3 ≤ 100
This states that five times a variable k, minus 3, is less than or equal to 100.
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Key Insights
Step 2: Add 3 to Both Sides
To isolate the term containing k, add 3 to both sides:
5k − 3 + 3 ≤ 100 + 3
Simplifies to:
5k ≤ 103
This step maintains balance by performing the same operation on both sides.
Step 3: Divide Both Sides by 5
Next, divide both sides by 5 to solve for k:
(5k)/5 ≤ 103/5
Which simplifies to:
k ≤ 103/5
Now, compute the division:
103 ÷ 5 = 20.6
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Thus, the final result is:
k ≤ 20.6
Why This Inequality Matters
Understanding and manipulating inequalities such as 5k − 3 ≤ 100 is valuable in fields like finance, engineering, and computer science. For example:
- Budgeting: If k represents units of a product, and total cost constraints depend linearly, this inequality helps determine maximum allowable units.
- Optimization: Engineers often use inequalities to define limits within which systems operate safely.
- Programming: Algorithms frequently evaluate inequality conditions to control logic flow.
Recap of Key Transformations
| Step | Operation | Result |
|-------|-------------------------------|--------------------------|
| 1 | Start: 5k − 3 ≤ 100 | |
| 2 | Add 3: 5k ≤ 103 | |
| 3 | Divide by 5: k ≤ 20.6 | |
Final Notes
Solving linear inequalities methodically ensures accuracy, especially when translating mathematical conditions into practical limits. Remember, dividing by a positive number (like 5) preserves the inequality direction, while dividing by a negative would reverse it— always vigilant about sign changes.
Whether you're solving equations for homework, modeling systems in science, or building algorithms, mastering inequalities like 5k − 3 ≤ 100 ⇒ k ≤ 20.6 strengthens your analytical toolkit.
Key Takeaways:
- Always isolate the variable term before solving for k.
- Maintain inequality direction during operations.
- Express results as exact fractions or decimals for clarity.
- Apply this logic to real-world constraints across disciplines.
