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Chapter 2: Problem 53

Graph each inequality. Then write the solutions in interval notation. $$ 0 \leq y<2 $$

Short Answer

Expert verified
The solution is \([0, 2)\) on the number line.

Step by step solution

01

Understanding the Inequality

The inequality given is \( 0 \leq y < 2 \), which means that \( y \) can be any value between \( 0 \) and \( 2 \), including \( 0 \) but not including \( 2 \).
02

Graph the Inequality

To graph \( 0 \leq y < 2 \), draw a number line. Place a filled circle (鈼�) at \( y = 0 \) to represent inclusion and an open circle (鈼�) at \( y = 2 \) to represent exclusion. Shade the region between \( 0 \) and \( 2 \) to show all the possible values of \( y \).
03

Express in Interval Notation

The interval notation for \( 0 \leq y < 2 \) is \( [0, 2) \). The square bracket [ indicates inclusion of \( 0 \), and the parenthesis ) indicates exclusion of \( 2 \).

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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 succinct way to describe a range of values on a number line. It conveys which numbers are included in a particular set or interval. In the case of the inequality \(0 \leq y < 2\), interval notation is represented as \([0, 2)\). In this notation:
  • The square bracket \([\) means that the value is included; so, \(0\) is part of the solution set.
  • The parenthesis \()\) means that the value is not included; so, \(2\) is excluded from the solution set.
This notation is both concise and clear, making it easy to understand at a glance which numbers are part of the solution.
Number Line Graphing
Graphing an inequality on a number line visually shows which values satisfy the inequality. To represent \(0 \leq y < 2\):
  • Begin by drawing a horizontal line, which is your number line.
  • Mark the critical points, \(0\) and \(2\), on this line.
  • At \(0\), place a filled circle (鈼�) to indicate that this value is included in the solution.
  • At \(2\), place an open circle (鈼�) to indicate that this value is not included.
  • Shade the region between these two points to demonstrate all possible values of \(y\) that satisfy the inequality.
This graph is a straightforward visual tool that helps confirm and understand the solution described by the inequality.
Graphing Inequalities
Graphing inequalities involves representing ranges on number lines or graphs to visualize solutions. Here are some basic steps for graphing linear inequalities like \(0 \leq y < 2\):
  • Identify the endpoint values of the inequality. In this example, these are \(0\) and \(2\).
  • Determine whether each endpoint is included or excluded in the solution set. This is based on the inequality symbols: \(\leq\) or \(<\).
  • Graphically represent included endpoints with filled circles and excluded endpoints with open circles.
  • Shade the region that falls between (or beyond, depending on the inequality) the endpoints, representing all the solutions that the inequality covers.
This method of graphing inequalities is a simple and effective way to make the solution set clear and understandable. This approach helps learners visually grasp how inequalities define ranges of potential values on a number line.

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