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Chapter 0: Problem 1

Rewrite in interval notation and graph. $$-2 \leq x < 3$$

Short Answer

Expert verified
In interval notation: \([-2, 3)\). Graph: solid dot at -2, open dot at 3, shade between.

Step by step solution

01

Understand the Inequality

The given inequality is \[-2 \leq x < 3.\]This expression means that \(x\) can take any value from \(-2\) to \(3\), including \(-2\) but not including \(3\).
02

Rewrite in Interval Notation

In interval notation, brackets are used to indicate whether endpoints are included or excluded. A square bracket \([\) means the endpoint is included, and a parenthesis \((\) means the endpoint is excluded. Thus, the inequality \[-2 \leq x < 3\]can be written in interval notation as \[[-2, 3).\]
03

Graph the Interval

On a number line, draw a solid dot at \(-2\) to denote that it is included in the interval and an open dot at \(3\) to denote that \(3\) is not included in the interval. Shade the region between \(-2\) and \(3\) to illustrate all the values \(x\) can take.

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Key Concepts

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

Inequalities
In the world of mathematics, inequalities are expressions that define a range of possible values for a variable. Essentially, they tell us where the value could lie on a scale. In the given exercise, we are working with the inequality \(-2 \leq x < 3\). This expression is a way to say that \(x\) can be any number that is greater than or equal to \-2\, and less than \(3\), as follows:
  • \(x\) is greater than or equal to \-2\, represented as \-2 \leq x\.
  • \(x\) is less than 3, represented as \(x < 3\).
Inequalities are different from equations because they do not represent exact values but instead a range of possibilities. This range can include all numbers within certain limits. Using inequalities helps in understanding scenarios where variables can vary within a spectrum of values.
Graphing Inequalities
Graphing inequalities on a number line is a visual way to present all possible solutions of an inequality. When graphing the inequality \(-2 \leq x < 3\), we need to portray which numbers \(x\) can actually be:
  • For \(-2\), since it is 'less than or equal to', we use a solid dot.
  • For \(3\), because \(x\) is 'less than', we use an open dot.
The solid dot at -2 shows inclusion, which means that \-2\ is part of the solution. Then, shading the region between \-2\ and \(3\) indicates all numbers in this interval are possible solutions. The open dot at \(3\) signifies it is not part of the solution range. This clear visual representation helps in quickly identifying the limits of the inequality's solution.
Number Line Representation
Number line representation is a straightforward way to visualize the range of numbers defined by an inequality. It uses dots and shading to show which numbers are included in the set of solutions. For our specific inequality \(-2 \leq x < 3\):
  • Draw a line segment on the number line.
  • At \(-2\), place a solid dot to indicate it is included.
  • Shade the segment between \-2\ and \3\ to show the range of values that \(x\) can take.
  • At \(3\), place an open dot to show it is excluded from the interval.
This method ensures anyone looking at the number line can easily understand the allowed values in the inequality. Number line graphs are used widely in algebra to communicate different types of inequalities and their solutions effectively.

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