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Chapter 3: Problem 8

For what positive values of \(x\) will \(x^{20}\) be greater than \(x^{18} ?\)

Short Answer

Expert verified
The positive values of \(x\) that satisfy \(x^{20} > x^{18}\) are \(x > 1\).

Step by step solution

01

Understand the Problem

We need to find the positive values of \(x\) for which \(x^{20} > x^{18}\).
02

Rearrange the Inequality

Divide both sides of the inequality \(x^{20} > x^{18}\) by \(x^{18}\) to simplify it.
03

Simplify the Inequality

When we divide both sides by \(x^{18}\), we get \(x^2 > 1\).
04

Solve the Simplified Inequality

To solve \(x^2 > 1\), take the square root of both sides. This gives us two cases: \(x > 1\) and \(x < -1\). However, since we only want positive values, we consider only the case where \(x > 1\).
05

Conclusion

The solution to the inequality \(x^{20} > x^{18}\) for positive values of \(x\) is \(x > 1\) .

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Key Concepts

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

Understanding Algebraic Inequalities
An algebraic inequality is a mathematical statement that compares two expressions with an inequality symbol like > or <. In our exercise, we are comparing powers of the same variable, specifically looking for values of x where one power is greater than another.

Steps to handle algebraic inequalities generally involve:
  • Rearranging the inequality
  • Simplifying the expressions
  • Solving the simplified inequality
This method helps isolate the variable we are solving for, making it easier to identify the required range of values.
Working with Exponents
Exponents express how many times a number, called the base, is multiplied by itself. In our inequality \(x^{20} > x^{18}\), we are comparing two expressions with the same base but different exponents.

The key steps for solving inequalities involving exponents include:
  • Performing operations that simplify the bases
  • Using properties of exponents, such as reducing the inequality by dividing by the lesser exponent
For example, in our problem, dividing both sides by \(x^{18}\) reduces \(x^{20}\) to \(x^2\), making the comparison simpler.
Positive Values and Their Importance
In math, dealing with positive values is crucial as they avoid negative results that may not make sense in context. For this particular exercise, we are asked to find only the positive values of x that satisfy the inequality.

After simplifying \(x^{20} > x^{18}\) to \(x^2 > 1\), it鈥檚 essential to consider only the positive solution to \(x^2 > 1\). While mathematical solutions can include negative values, we limit ourselves to positives, meaning we consider only \(x > 1\) as a valid solution, consistent with the problem's requirements.

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Most popular questions from this chapter

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