Five Less Than A Number Is Greater Than Twenty

Five less than a number is greater than twenty—this seemingly simple statement opens a door to the fascinating world of inequalities and mathematical problem-solving. Understanding how to translate these words into a mathematical expression and then solve it is a fundamental skill in algebra. This article will meticulously break down the process, ensuring you grasp not only the mechanics but also the underlying logic.

Decoding the Phrase: From Words to Symbols

The first step in tackling "five less than a number is greater than twenty" is to translate it into a mathematical inequality. Let's dissect the phrase piece by piece:

• "A number": This refers to an unknown value, which we typically represent with a variable. Let's use 'x' to denote this unknown number.
• "Five less than a number": This means we are subtracting 5 from the number 'x'. So, this part translates to 'x - 5'.
• "Is greater than twenty": This establishes the relationship between 'x - 5' and 20. The phrase "is greater than" is represented by the inequality symbol '>'. Thus, this translates to '> 20'.

Putting it all together, we get the inequality: x - 5 > 20.

Solving the Inequality: Unveiling the Possible Values

Now that we have our mathematical expression, x - 5 > 20, we can proceed to solve it. Solving an inequality is similar to solving an equation, with one crucial difference: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. However, in this case, we don't need to worry about that.

Here's how we solve the inequality:

1. Isolate the variable: Our goal is to get 'x' by itself on one side of the inequality. To do this, we need to get rid of the '- 5' on the left side.
2. Add 5 to both sides: To eliminate the '- 5', we perform the inverse operation, which is adding 5. We must add 5 to both sides of the inequality to maintain the balance.
   • x - 5 + 5 > 20 + 5
3. Simplify: After adding 5 to both sides, we simplify:
   • x > 25

Therefore, the solution to the inequality x - 5 > 20 is x > 25. This means that any number greater than 25 will satisfy the original statement "five less than a number is greater than twenty."

Understanding the Solution: A Range of Possibilities

The solution x > 25 isn't just a single number; it represents a range of numbers. It signifies that any number greater than 25 will make the original statement true. For example:

• If x = 26, then 26 - 5 = 21, which is greater than 20.
• If x = 30, then 30 - 5 = 25, which is not greater than 20 (it's equal to, which we'll discuss later).
• If x = 100, then 100 - 5 = 95, which is greater than 20.

This illustrates that the solution x > 25 encompasses an infinite number of values.

Visualizing the Solution: The Number Line

A number line provides a visual representation of the solution to an inequality. To represent x > 25 on a number line:

1. Draw a number line: Draw a horizontal line and mark points representing numbers, including 25.
2. Mark the critical value: Place an open circle at 25. This open circle indicates that 25 is not included in the solution set because the inequality is strictly "greater than" (x > 25). If the inequality were "greater than or equal to" (x ≥ 25), we would use a closed circle to indicate that 25 is included.
3. Shade the solution set: Shade the portion of the number line to the right of 25. This shaded area represents all the numbers that are greater than 25.
4. Draw an arrow: Draw an arrow extending to the right, indicating that the solution continues infinitely in the positive direction.

The number line visually reinforces that the solution to x > 25 is not a single point but a continuous range of values.

Expanding the Concept: "Greater Than or Equal To"

The phrase "is greater than or equal to" introduces a slight but important variation. In mathematics, this is represented by the symbol '≥'. Let's consider the statement: "Five less than a number is greater than or equal to twenty."

1. Translate: Following the same logic as before, we translate this into the inequality: x - 5 ≥ 20.
2. Solve: We solve this inequality in the same way:
   • x - 5 + 5 ≥ 20 + 5
   • x ≥ 25

The solution x ≥ 25 means that any number greater than or equal to 25 will satisfy the original statement. This includes 25 itself.

On a number line, the representation of x ≥ 25 would differ slightly from x > 25. Instead of an open circle at 25, we would use a closed circle to indicate that 25 is included in the solution set. The rest of the number line would be shaded to the right, just like before.

Real-World Applications: Bringing Inequalities to Life

Inequalities aren't just abstract mathematical concepts; they have practical applications in numerous real-world scenarios. Here are a few examples:

• Minimum Requirements: Imagine a roller coaster with the requirement: "You must be at least 48 inches tall to ride." This can be represented as h ≥ 48, where 'h' is your height in inches.
• Budget Constraints: Suppose you have a budget of $50 for groceries. If 'x' represents the amount you spend, the inequality would be x ≤ 50 (x is less than or equal to 50).
• Speed Limits: A speed limit of 65 mph can be represented as s ≤ 65, where 's' is your speed.
• Profit Margins: A company might require a profit margin of at least 20%. If 'p' represents the profit margin, then p ≥ 20.
• Exam Scores: To get an A in a class, you might need a score of at least 90. If 's' represents your score, then s ≥ 90.

These examples demonstrate how inequalities are used to express constraints, limits, and minimum or maximum values in various contexts.

Combining Inequalities: Compound Inequalities

Sometimes, we need to express a range of values between two limits. This is where compound inequalities come in. A compound inequality combines two inequalities using "and" or "or."

• "And" Inequalities: These represent values that satisfy both inequalities simultaneously. For example, "x is greater than 5 and less than 10" can be written as 5 < x < 10. This means x must be both greater than 5 and less than 10. On a number line, this would be represented by a shaded line segment between 5 and 10, with open circles at both endpoints (unless the inequalities were "greater than or equal to" or "less than or equal to").
• "Or" Inequalities: These represent values that satisfy either one inequality or the other (or both). For example, "x is less than 2 or greater than 6" can be written as x < 2 or x > 6. This means x can be either less than 2 or greater than 6. On a number line, this would be represented by two separate shaded regions: one to the left of 2 and one to the right of 6.

Advanced Scenarios: Multi-Step Inequalities

Just like equations, inequalities can involve multiple steps to solve. These multi-step inequalities require careful application of the same principles we've already discussed, combined with the order of operations.

For example, consider the inequality: 2(x + 3) - 5 > 7

To solve this:

1. Distribute: First, distribute the 2:
   • 2x + 6 - 5 > 7
2. Combine Like Terms: Combine the constants:
   • 2x + 1 > 7
3. Isolate the Variable: Subtract 1 from both sides:
   • 2x > 6
4. Divide: Divide both sides by 2:
   • x > 3

Therefore, the solution to the inequality is x > 3.

Special Cases: All Real Numbers and No Solution

Not all inequalities have a straightforward solution like x > 25. There are two special cases to be aware of:

• All Real Numbers: Sometimes, after simplifying an inequality, you might end up with a statement that is always true, regardless of the value of 'x'. For example: x + 1 > x. Subtracting 'x' from both sides leaves 1 > 0, which is always true. In this case, the solution is "all real numbers," meaning any value of 'x' will satisfy the original inequality.
• No Solution: Conversely, you might end up with a statement that is always false. For example: x < x - 1. Subtracting 'x' from both sides leaves 0 < -1, which is never true. In this case, the solution is "no solution," meaning there is no value of 'x' that will satisfy the original inequality.

Common Mistakes to Avoid: Staying on Track

When working with inequalities, it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

• Forgetting to Reverse the Inequality Sign: The most common mistake is forgetting to reverse the direction of the inequality sign when multiplying or dividing both sides by a negative number. This is a critical rule to remember.
• Incorrect Distribution: Make sure to distribute correctly when dealing with parentheses. Multiply the term outside the parentheses by every term inside.
• Combining Unlike Terms: Only combine terms that have the same variable and exponent. For example, you can combine 2x and 3x, but you cannot combine 2x and 3x².
• Misinterpreting the Inequality Symbols: Understand the difference between >, <, ≥, and ≤. Remember that ≥ and ≤ include the possibility of equality.
• Ignoring the Number Line: Using a number line can help you visualize the solution set and avoid errors.
• Not Checking Your Solution: After solving an inequality, it's always a good idea to check your solution by plugging in a value from your solution set back into the original inequality to make sure it holds true.

Conclusion: Mastering the Art of Inequalities

The statement "five less than a number is greater than twenty" is a simple entry point into the world of inequalities. By understanding how to translate these words into mathematical expressions, solve them using algebraic techniques, and interpret the solutions both numerically and visually, you can build a strong foundation for more advanced mathematical concepts. Remember to pay attention to detail, avoid common mistakes, and practice regularly to master the art of inequalities. Inequalities are not just abstract symbols; they are powerful tools for representing and solving real-world problems, from setting budget constraints to determining minimum requirements.