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Chapter 9: Problem 45

Explain why \(|3 t-7|<0\) has no solution.

Short Answer

Expert verified
The given inequality \(|3t-7|<0\) contradicts the basic properties of absolute values, which state that the absolute value of any real number is always non-negative (i.e., \(|a|\ge 0\) for any real number \(a\)). Therefore, there is no real number \(t\) that can satisfy the inequality \(|3t-7|<0\), which implies that it has no solution.

Step by step solution

01

Understand the properties of absolute value

The absolute value of a real number is its distance from zero. This distance is always non-negative, meaning it's either positive or zero. Mathematically, this can be represented as: \(|a| \geq 0\), where \(a\) is any real number.
02

Analyze the given inequality

We are given the inequality \(|3t-7| < 0\). Notice that this inequality contradicts the properties of absolute value, which states that the absolute value of a real number is always non-negative. Therefore, the inequality \(|3t-7| < 0\) can never be true.
03

Make a conclusion

Since the inequality contradicts the basic properties of absolute values, it implies that there is no real number \(t\) that can satisfy the inequality \(|3t-7| < 0\). Therefore, the given inequality has no solution.

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