A Number Minus 3 Is At Least -5.

A Number Minus 3 is at Least -5: Understanding and Solving Inequalities

This article delves into the meaning and solution of the inequality "a number minus 3 is at least -5." We'll explore the mathematical concepts behind this statement, break down the steps to solve it, and examine its applications in real-world scenarios. Understanding inequalities is crucial for various fields, from basic algebra to advanced calculus and beyond. This seemingly simple inequality provides a gateway to grasping more complex mathematical concepts.

Understanding the Problem: Deconstructing the Inequality

The phrase "a number minus 3 is at least -5" translates directly into a mathematical inequality. Let's break it down:

• "a number": This represents an unknown value, typically denoted by a variable, such as x.
• "minus 3": This indicates subtraction; we subtract 3 from our unknown number.
• "is at least": This is the key phrase that signifies an inequality. "At least" means the result is greater than or equal to -5. This is represented by the symbol ≥.
• "-5": This is the lower limit or boundary of the inequality.

Therefore, the complete mathematical representation of the sentence is: x - 3 ≥ -5

Steps to Solve the Inequality: Isolating the Variable

Solving an inequality involves isolating the variable (x in this case) to determine its possible values. The process is similar to solving an equation, but with one crucial difference: when multiplying or dividing by a negative number, the inequality sign flips. Let's solve our inequality step-by-step:

1. Add 3 to both sides: To isolate x, we need to undo the subtraction of 3. We achieve this by adding 3 to both sides of the inequality:

   x - 3 + 3 ≥ -5 + 3

2. Simplify: This simplifies the inequality to:

   x ≥ -2

This solution tells us that x can be any number greater than or equal to -2.

Representing the Solution: Visualizing the Inequality

The solution, x ≥ -2, can be represented graphically on a number line and using interval notation.

• Number Line: Draw a number line. Mark -2 on the line. Since x can be equal to -2, we use a closed circle (or a filled-in dot) at -2. Then, shade the region to the right of -2, indicating all values greater than -2 are included in the solution set.

• Interval Notation: Interval notation is a concise way to represent a range of numbers. In this case, the solution is represented as [-2, ∞). The square bracket "[" indicates that -2 is included, and the parenthesis ")" indicates that infinity (∞) is not included (as infinity is a concept, not a specific number).

Deeper Dive: Understanding the Concepts

Let's delve deeper into the underlying mathematical concepts:

• Inequalities vs. Equations: While solving inequalities shares similarities with solving equations, there's a fundamental difference. Equations (e.g., x - 3 = -2) have a single solution, while inequalities (e.g., x - 3 ≥ -5) have a range of solutions.

• The Importance of the Inequality Sign: The inequality sign (≥ in this case) dictates the direction of the solution. If the sign were < (less than), the solution and graphical representation would be different.

• Real-World Applications: Inequalities are frequently used to model real-world situations involving constraints or limitations. For example, this inequality could represent a minimum profit requirement for a business, a minimum score needed on a test, or a minimum temperature requirement for a process.

• Compound Inequalities: More complex problems might involve compound inequalities, combining multiple inequalities. For instance, a scenario might require a number to be both greater than -2 and less than 5, represented as -2 ≤ x < 5.

Illustrative Examples: Applying the Concepts

Let's explore a few examples to solidify our understanding:

Example 1: Suppose a certain product needs to maintain a minimum temperature of -5 degrees Celsius. If the current temperature is 3 degrees less than the ideal temperature, what is the minimum acceptable current temperature?

This translates to the inequality x - 3 ≥ -5, where x represents the current temperature. Solving this inequality yields x ≥ -2, meaning the minimum acceptable current temperature is -2 degrees Celsius.

Example 2: A student needs to score at least 80% on a final exam to pass the course. If their current average is 3% less than the passing grade, what minimum score do they need on the final exam?

This is represented as x - 3 ≥ 80. Solving this, we get x ≥ 83. The student must score at least 83% on the final exam.

Example 3: A company requires a minimum profit of $10,000 per month. If their current profit is $3,000 less than their target, what is the minimum acceptable current profit?

This scenario can be represented by the inequality x - 3000 ≥ 10000. Solving for x, we get x ≥ 13000. The company must earn at least $13,000 in profit this month.

Frequently Asked Questions (FAQ)

Q: What happens if the inequality sign is reversed?

A: If the inequality were x - 3 ≤ -5, the solution process would be the same, but the solution would be x ≤ -2. This means x can be any number less than or equal to -2. The inequality sign dictates the direction of the solution on the number line and in interval notation.

Q: Can I multiply both sides of the inequality by a negative number?

A: Yes, but you must remember to reverse the inequality sign. For example, if you have -x ≥ 2, multiplying both sides by -1 gives x ≤ -2. Forgetting to reverse the inequality sign is a common mistake.

Q: What if the inequality involves more than one operation?

A: Follow the order of operations (PEMDAS/BODMAS) and apply the same principles of adding or subtracting to both sides and reversing the inequality sign when multiplying or dividing by a negative number. For instance, if you have 2x + 5 ≥ 11, you would first subtract 5 from both sides, then divide by 2.

Q: How can I check my solution?

A: To verify your solution, choose a value within your solution set and plug it back into the original inequality. If the inequality holds true, your solution is correct. For example, for x ≥ -2, you could test x = 0: 0 - 3 ≥ -5, which simplifies to -3 ≥ -5 (true).

Conclusion: Mastering Inequalities

The inequality "a number minus 3 is at least -5" may seem simple at first glance, but it serves as a foundational concept for understanding and solving more complex inequalities. By mastering the steps involved in solving inequalities and understanding their graphical and interval representations, you equip yourself with a powerful tool applicable across numerous mathematical and real-world contexts. Remember the crucial rule of reversing the inequality sign when multiplying or dividing by a negative number, and always check your solutions to ensure accuracy. With practice, solving inequalities will become second nature, empowering you to confidently tackle more advanced mathematical challenges.