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Chapter 3: Problem 48

For this problem, assume \(x\) is positive. a. Name three values of \(x\) for which \(\sqrt{x}\) is less than \(x\) . b. Name three values of \(x\) for which \(\sqrt{x}\) is greater than \(x\) . c. In general, how can you tell whether \(\sqrt{x}\) is greater than \(x ?\)

Short Answer

Expert verified
For \(\sqrt{x} < x\), choose values like 4, 9, and 16. For \(\sqrt{x} > x\), choose values like \(\frac{1}{4}\), \(\frac{1}{9}\), and \(\frac{1}{16}\). Generally, \(\sqrt{x} < x\) if \(x > 1\) and \(\sqrt{x} > x\) if \(0 < x < 1\).

Step by step solution

01

- Solving part (a) for \sqrt{x} < x

To find values of \(x\) for which \(\sqrt{x} < x\), consider when \(x > 1\). For example, evaluate \(x = 4\), \(x = 9\), and \(x = 16\). Verify each: \(\sqrt{4} = 2\), and \(2 < 4\); \(\sqrt{9} = 3\), and \(3 < 9\); \(\sqrt{16} = 4\), and \(4 < 16\).
02

- Solving part (b) for \sqrt{x} > x

To find values of \(x\) for which \(\sqrt{x} > x\), consider when \(0 < x < 1\). For example, evaluate \(x = \frac{1}{4}\), \(x = \frac{1}{9}\), and \(x = \frac{1}{16}\). Verify each: \(\sqrt{\frac{1}{4}} = \frac{1}{2}\), and \(\frac{1}{2} > \frac{1}{4}\); \(\sqrt{\frac{1}{9}} = \frac{1}{3}\), and \(\frac{1}{3} > \frac{1}{9}\); \(\sqrt{\frac{1}{16}} = \frac{1}{4}\), and \(\frac{1}{4} > \frac{1}{16}\).
03

- Solving part (c) for general rule

In general, \(x\) determines if \(\sqrt{x} \) is greater or less than \(x\) by the interval of \(x\). If \(x > 1\), then \(\sqrt{x} < x\). If \(0 < x < 1\), then \(\sqrt{x} > x\).

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Key Concepts

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

Square Root Properties
When dealing with square roots, there are a few key properties to keep in mind.
A square root of a number is a value that, when multiplied by itself, gives the original number. Mathematically, the square root of a number x is denoted as \(\sqrt{x}\)\.
Here are some fundamental properties:
  • \(\sqrt{x^2} = |x|\).
  • \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \text{ for positive } a \text{ and } b \).
  • \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \text{ where } b eq 0 \text{} \).
Property one reminds us that the square root function always yields a non-negative result.
These properties are foundational when solving algebraic inequalities involving square roots.
Algebraic Inequalities
Algebraic inequalities are expressions that use inequality signs instead of equality signs.
Common inequality signs include:
  • \(>\): greater than
  • \(<\): less than
  • \(\geq\): greater than or equal to
  • \(\leq\): less than or equal to
When solving inequalities, the goal is to find the set of values for a variable that make the inequality true.
For example, determining when \(\sqrt{x}\) is less than or greater than \(x\) involves understanding the intervals of x where this is true.
There are specific methods for handling inequalities involving square roots:
  • When \(x > 1\), \(\sqrt{x} < x\).
  • When \(0 < x < 1\), \(\sqrt{x} > x\).
Understanding these relationships helps in determining solutions to problems that involve both square roots and inequalities.
Solving Inequalities
Solving inequalities is similar to solving equations, but with additional considerations for inequality signs.
It's crucial to ensure that the steps taken preserve the direction of the inequality.
Here's a general approach:
  1. Isolate the variable on one side of the inequality.
  2. Use properties of inequalities to manipulate the expressions. Be careful with multiplication or division by a negative number, as this reverses the inequality sign.
  3. Check interval conditions where the inequality holds true.
For example, to determine when \(\sqrt{x} < x\), you can test values greater than 1:
  • For \(x = 4\), \(\sqrt{4} = 2\), and \(2 < 4\).
  • For \(x = 9\), \(\sqrt{9} = 3\), and \(3 < 9\).
  • For \(x = 16\), \(\sqrt{16} = 4\), and \(4 < 16\).
When \(0 < x < 1 \), you find \(\sqrt{x} > x\):
  • For \(x = \frac{1}{4}\), \(\sqrt{\frac{1}{4}} = \frac{1}{2}\), and \(\frac{1}{2} > \frac{1}{4}\).
  • For \(x = \frac{1}{9}\), \(\sqrt{\frac{1}{9}} = \frac{1}{3}\), and \(\frac{1}{3} > \frac{1}{9}\).
  • For \(x = \frac{1}{16}\), \(\sqrt{\frac{1}{16}} = \frac{1}{4}\), and \(\frac{1}{4} > \frac{1}{16}\).
Following these steps helps to accurately solve inequalities involving square roots.

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