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Chapter 0: Problem 12

Use intervals to describe the real numbers satisfying the inequalities. $$x \geq \sqrt{2}$$

Short Answer

Expert verified
[\sqrt{2}, \infty)\].

Step by step solution

01

Identify the Inequality

The given inequality is: \[x \geq \sqrt{2}\.\]
02

Understand the Meaning of the Inequality

This inequality means that the value of \(x\) can be equal to or greater than \(\sqrt{2}\).
03

Determine the Interval

Since \(x\) is greater than or equal to \(\sqrt{2}\), the interval starts at \(\sqrt{2}\) and extends to positive infinity.
04

Write the Interval Notation

In interval notation, the real numbers satisfying the inequality \(x \geq \sqrt{2}\) are written as \[ [\sqrt{2} , \infty) \]

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Key Concepts

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

Understanding Inequalities
Inequalities tell us about the relative size or order of two values. When working with inequalities, you might encounter symbols like <, >, 鈮�, or 鈮�. These symbols help us understand which values are greater or smaller. For example, the inequality \(x \geq \sqrt{2}\) says that \(x\) is either bigger than or equal to the square root of 2. Inequalities can be solved to find ranges of values, and these ranges can be expressed using interval notation.
What Are Real Numbers?
Real numbers include all the numbers you normally use in everyday life, like whole numbers, fractions, and decimals. Anything that can be placed on a number line falls under real numbers. For example, 3, -2, 0.5, and \( \sqrt{2}\) are all real numbers. When solving inequalities, you are typically looking for a range of real numbers that can satisfy a given condition. In our case, we want all real numbers greater than or equal to \( \sqrt{2}\).
Introduction to Intervals
Intervals are a way to describe a range of numbers. They help us understand which numbers fit into a certain set. There are a few types of intervals:
  • Closed Interval [a, b]: Includes both endpoints a and b.
  • Open Interval (a, b): Excludes both endpoints a and b.
  • Half-Open Interval [a, b) or (a, b]: Includes one endpoint but not the other.
To express the solution to \(x \geq \sqrt{2} \), we use intervals. Our interval starts at \(\sqrt{2}\) (included) and goes to infinity (which we can't ever reach, so it's not included). Thus, we write \([\sqrt{2}, \infty ) \).

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