Problem 61 - Prealgebra

Question

For what value of \(x\) is \(\frac{1}{\sqrt{x}}<\sqrt{x}

Step-by-Step Solution

Answer

Option D, 2, satisfies the inequality.

Step 1: Understanding the Inequality

The inequality \( \frac{1}{\sqrt{x}} < \sqrt{x} < x \) is given, and it needs to be solved for \( x \). This involves establishing a range of values for which this inequality holds true.

Step 2: Solve the First Part of the Inequality

We need to solve \( \frac{1}{\sqrt{x}} < \sqrt{x} \). By simplifying this, we get \( 1 < x \). The values of \( x \) must be greater than 1 for this inequality to hold.

Step 3: Solve the Second Part of the Inequality

Next, solve \( \sqrt{x} < x \). Squaring both sides, we have \( x < x^2 \). Simplifying gives \( 1 < x \) if \( x > 1 \). The solution is consistent with \( x > 1 \).

Step 4: Combine Results

Both inequalities \( 1 < x \) from \( \frac{1}{\sqrt{x}} < \sqrt{x} \) and \( x > 1 \) from \( \sqrt{x} < x \) suggest that \( x > 1 \). Thus, \( x \) must be greater than 1 for both inequalities to hold.

Step 5: Check the Options

Check option values to see which are greater than 1. Only option D, 2, satisfies \( x > 1 \). Substituting \( x = 2 \) into the original inequalities: \( \frac{1}{\sqrt{2}} < \sqrt{2} < 2 \) holds true.

Key Concepts

Understanding Inequalities

Inequalities are expressions where two values are compared with operators like less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). They form the foundation of many mathematical concepts and are used to show how numbers relate to one another.
When solving inequalities, our goal is to find the range of values that satisfy the condition. For example, in the inequality shown in the exercise, \( \frac{1}{\sqrt{x}} < \sqrt{x} < x \), the challenge is to discover for which values of \( x \) all these conditions are true simultaneously.

• The first inequality, \( \frac{1}{\sqrt{x}} < \sqrt{x} \), suggests that the reciprocal of one square root should be less than another square root.
• The second part, \( \sqrt{x} < x \), tells us that the square root of \( x \) should be less than \( x \).

Both parts must be satisfied at the same time by the same value of \( x \), making inequalities a powerful tool for understanding these relationships.

Exploring Square Roots

Square roots are fundamental to the study of mathematics. A square root of a number \( y \) is a value \( x \) such that when multiplied by itself, the result is \( y \); in other words, \( x^2 = y \).
Square roots are represented by the symbol \( \sqrt{ } \) and have important properties that make them unique, such as the relationship between square roots and exponents.

• A positive number has two square roots: a positive and a negative one. However, in most situations, we consider only the principal (positive) square root.
• The square root helps us simplify radical expressions and solve equations, making it a versatile mathematical tool.

In the context of the exercise, bringing in the notion of square roots allows us to transition from a complex inequality into something that is easier to evaluate, such as comparing the roots directly or their operations. This transformation is crucial to solving complex inequality problems effectively.

Reasoning Through Step-by-Step Solutions

Approaching mathematical problems in a step-by-step manner helps clarify the process and reduces errors in computation or logic. By breaking down each component into smaller parts, the problem often becomes easier to manage and resolve.

Let's briefly outline the step-by-step method used in solving the given inequality:

• Firstly, identify the inequality \(( \frac{1}{\sqrt{x}} < \sqrt{x} < x )\) and interpret what it's asking us to find.
• Secondly, tackle each part of the inequality individually. Start with \( \frac{1}{\sqrt{x}} < \sqrt{x} \), simplify it, and find that it implies \( 1 < x \).
• Next, address \( \sqrt{x} < x \), which after simplification also results in \( 1 < x \).
• Finally, combine the results and check the values within the options provided to see which fits all parts of the inequality.

This methodical process ends with a precise determination of the solution, reinforcing the importance of clear reasoning and logical steps in problem-solving. It encourages both understanding and confidence in handling inequalities and similar mathematical challenges.