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Chapter 1: Problem 51

Express the interval in terms of inequalities, and then graph the interval. $$[2, \infty)$$

Short Answer

Expert verified
The interval \([2, \infty)\) is expressed as \(x \geq 2\).

Step by step solution

01

Identify the interval type

The interval \([2, \infty)\) is a half-open interval. It starts at 2 and includes that value while extending to infinity without including it.
02

Express the interval as an inequality

For the interval \([2, \infty)\), we need to express it as an inequality. Since the interval starts at 2 and includes 2, it translates to \(x \geq 2\). As the interval extends to infinity, it doesn't impose an upper limit on \(x\). Thus, the inequality is \(x \geq 2\).
03

Graph the interval

To graph \([2, \infty)\), draw a number line. Place a solid dot at 2 to indicate that 2 is included. Then, draw an arrow extending to the right from 2, which continues indefinitely to represent extending to infinity.

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Key Concepts

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

Interval Notation
Interval notation is a concise way of expressing subsets of real numbers.
This notation uses brackets and parentheses to describe the set of numbers that satisfy an inequality.

Consider the interval \([2, \infty)\).
  • The square bracket \([\) indicates the inclusion of the boundary point, meaning 2 is included in the set.
  • The round parenthesis \()\) at infinity signifies that infinity is not a specific number, so it is never included in the interval.
This notation effectively condenses the inequality \(x \geq 2\) into a single expression, providing a clear visual representation of the range.
Graphing Intervals
Graphing intervals visually represents the range of values described by the interval notation.
It's especially useful in understanding and interpreting inequalities.

For the interval \([2, \infty)\):
  • Start by drawing a horizontal line to represent the number line.
  • Identify the starting point, 2, and mark it with a solid dot since 2 is included in the interval.
  • Draw an arrow starting from the solid dot and extending to the right, indicating the numbers greater than 2 and continuing indefinitely.
This graphical approach helps in recognizing that all numbers from 2 onwards are part of the solution set.
Number Line Representation
The number line is a fundamental tool in mathematics that helps in understanding the position and relationship of numbers.
It simplifies the visualization of an interval from a set of given solutions.

Using a number line for \([2, \infty)\):
  • The line itself extends infinitely in both positive and negative directions, but we focus on the relevant section according to the problem.
  • A solid dot on the number line at 2 means 2 itself is in the set.
  • The arrow pointing to the right suggests all numbers greater than 2 are included in the interval.
Incorporating the number line provides clarity in comprehending how inequality works through visual placement and direction.

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Most popular questions from this chapter

Fish Population The fish population in a certain lake rises and falls according to the formula $$F=1000\left(30+17 t-t^{2}\right)$$ Here \(F\) is the number of fish at time \(t,\) where \(t\) is measured in years since January \(1,2002,\) when the fish population was first estimated. (a) On what date will the fish population again be the same as it was on January \(1,2002 ?\) (b) By what date will all the fish in the lake have died?

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Write the number indicated in each statement in scientific notation. (a) A light-year, the distance that light travels in one year, is about \(5,900,000,000,000 \mathrm{mi}\) (b) The diameter of an electron is about \(0.0000000000004 \mathrm{cm}\) (c) A drop of water contains more than 33 billion billion molecules.

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