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Chapter 2: Problem 88

Write inequality in interval notation, and graph the interval. \(3 \leq x \leq 5\)

Short Answer

Expert verified
[3, 5]

Step by step solution

01

Understand the Inequality

The given inequality is \(3 \leq x \leq 5\). This means that the value of \(x\) is between 3 and 5, inclusive of 3 and 5.
02

Write the Interval Notation

In interval notation, inclusive inequalities are represented with square brackets. Therefore, the interval notation for the inequality \(3 \leq x \leq 5\) is \([3, 5]\).
03

Graph the Interval

To graph the interval \([3, 5]\), draw a number line. Mark the points 3 and 5 with closed circles to indicate that these points are included in the interval, then shade the region between these two points.

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Key Concepts

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

Inequalities
Inequalities are a fundamental concept in mathematics. They tell us about the relative size or order of two values. For example, the inequality \(3 \leq x \leq 5\) states that \(x\) is at least 3 and at most 5. This includes the values 3 and 5 themselves, as indicated by the 'less than or equal to' symbol (\( \leq \)).

Here are the main symbols used in inequalities:
  • \( < \): Less than
  • \( \leq \): Less than or equal to
  • \( > \): Greater than
  • \( \geq \): Greater than or equal to

Understanding these symbols is crucial for solving and interpreting inequalities in various mathematical contexts.
Number Line
A number line is a visual representation of numbers in order, usually drawn as a horizontal line. Each point on the line corresponds to a number. You can use a number line to visually show the solution to an inequality.

Here's how to use a number line to express inequalities:
  • Draw a horizontal line and mark evenly spaced points.
  • Label critical points, such as 3 and 5 in our example.
  • Closed circles (鈼�) are used for \( \leq \) and \( \geq \), showing that endpoints are included.
  • Open circles (鈼�) are used for \( < \) and \( > \), indicating endpoints are not included.

In our inequality \(3 \leq x \leq 5\), both 3 and 5 will have closed circles because the values are included.
Graphing Intervals
Graphing intervals on a number line provides a visual way to represent sets of numbers. Here's how to graph the interval \([3, 5]\):

  • First, draw a number line and mark key points, in this case, 3 and 5.
  • Place a closed circle (鈼�) at 3 because 3 is included.
  • Place another closed circle (鈼�) at 5 because 5 is also included.
  • Shade the region between 3 and 5 to show all values \(x\) can take.

This shaded region visually represents all possible values of \(x\) that satisfy the inequality \(3 \leq x \leq 5\).

Graphing intervals makes it easier to see and understand which numbers are included in a solution set, providing clarity to otherwise abstract mathematical expressions.

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