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Chapter 2: Problem 90

Write inequality in interval notation, and graph the interval. \(-3 \leq x<0\)

Short Answer

Expert verified
Interval notation: \([-3, 0)\), Graph includes [-3, 0) with shaded region and open circle at 0.

Step by step solution

01

Identify the inequality

The inequality given is \(-3 \leq x < 0\). This means that x is greater than or equal to -3, and less than 0.
02

Write the inequality in interval notation

In interval notation, the inequality \(-3 \leq x < 0\) is written as \([-3, 0)\). The square bracket \([\) indicates that -3 is included in the interval, and the parenthesis \( )\) indicates that 0 is not included.
03

Graph the interval

To graph the interval \([-3, 0)\), draw a number line and mark points at -3 and 0. Draw a closed circle at -3 to indicate that -3 is included, and draw an open circle at 0 to indicate that 0 is not included. Shade the region between -3 and 0.

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Key Concepts

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

inequality
An inequality is a mathematical statement that compares two values or expressions. Instead of saying that the values are exactly equal, an inequality shows that one value is larger or smaller than the other, or that one value is at least as large or as small as the other. For example, the inequality \( -3 \leq x < 0 \) means that \( x \) can be any number between -3 and 0. However, \( x \) can include -3 but cannot include 0. Recognizing this distinction is crucial for solving problems involving inequalities.
number line
A number line is a visual representation of all potential values that numbers can take. It helps to easily see the relationship between numbers and is particularly useful for graphing inequalities. A number line is usually a horizontal line with evenly spaced marks that represent numbers. When working with inequalities, these marks help to show the range of acceptable values for the variable of interest.
graphing intervals
Graphing intervals on a number line is a visual way to depict the set of solutions for an inequality. To graph the interval \( [-3, 0) \), you would:
  • Draw a number line.
  • Mark the points -3 and 0 on the line.
  • Place a closed circle at -3 to show that this value is included in the interval.
  • Place an open circle at 0 to show that this value is not included in the interval.
  • Shade the region between -3 and 0 to indicate that all numbers between -3 and 0 are possible values for \( x \).
This creates a clear visual guide that shows exactly which numbers satisfy the inequality.
closed circle
A closed circle is used on a number line to indicate that a particular value is included in the interval. For instance, in the interval \( [-3, 0) \), a closed circle at -3 means that -3 is a valid solution for \( x \). Whenever you see a closed circle, remember that it signifies that the endpoint is part of the solution set.
open circle
An open circle is drawn on a number line to show that a particular value is not included in the interval. In our example \( [-3, 0) \), the open circle at 0 indicates that 0 is not a valid solution for \( x \). This visual cue helps you to quickly see which values are excluded from the set of solutions.

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