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Chapter 1: Problem 100

Explain why \(8

Short Answer

Expert verified
No number can be both greater than 8 and less than 2 at the same time.

Step by step solution

01

Understand the inequality

The inequality given is \(8 < x < 2\). It means that the value of \(x\) must be greater than 8 and less than 2 simultaneously.
02

Analyze the range of values

A number cannot be both greater than 8 and less than 2 at the same time. These two conditions are contradictory.
03

Conclusion

Since no number can satisfy both conditions \(x > 8\) and \(x < 2\) simultaneously, there is no value of \(x\) that will make the inequality true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Contradictory Conditions
In the world of algebra, sometimes we come across inequalities that simply don't make sense. These are called 'contradictory conditions'. For example, consider the inequality 8 < x < 2.
This means that the value of x must be both greater than 8 and less than 2 at the same time.
When two conditions contradict each other in this way, they create an impossible scenario.
No number can be greater than 8 and less than 2 at the same time.
Understanding contradictory conditions is crucial because it helps us figure out when an inequality cannot be solved.
It's like trying to find a number that is both positive and negative simultaneously; it just can't happen.
Inequality Range
Inequalities define the range of possible values for a given variable. When you see something like 3 < x < 7, it tells you the acceptable range for x is between 3 and 7.
But what happens when the range is not logical, like with 8 < x < 2?
The 'range' here is impossible because there's no number that fits both being greater than 8 and less than 2.
Always check the range your inequality describes.
If it doesn't make sense, it's an indication that the inequality has no solution.
Knowing how to analyze and interpret these ranges will save you time and prevent confusion.
No Solution in Inequalities
Sometimes, inequalities have no solution. When we have contradictory conditions, as in 8 < x < 2,
we quickly realize that no single value of x can satisfy this inequality.
This situation is known as there being 'no solution'.
In algebra, identifying when an inequality has no solution is just as important as solving those that do have solutions.
If you encounter an impossible condition, it's okay to conclude: there is no solution.
Recognizing this early can help you avoid endless frustration and allow you to focus on other problems you can solve.

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Most popular questions from this chapter

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