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Chapter 0: Problem 4

Rewrite in interval notation and graph. $$0 < x < 6$$

Short Answer

Expert verified
Interval notation: (0, 6); open circles at 0 and 6 on a number line.

Step by step solution

01

Identify the Inequality Bounds

The given inequality is \(0 < x < 6\). This tells us that \(x\) is greater than 0 and less than 6. This means that the values of \(x\) are between 0 and 6, but do not include 0 or 6 themselves.
02

Convert to Interval Notation

In interval notation, we use parentheses \((a, b)\) to indicate that the end values are not included (open interval). For the inequality \(0 < x < 6\), we write this as the interval \((0, 6)\). The parentheses indicate that 0 and 6 are not included.
03

Graph the Interval

To graph the interval \((0, 6)\), draw a number line. Place an open circle at 0 and another open circle at 6 to indicate that these points are not included. Then, draw a line connecting these two circles to show the values between them are part of the solution set.

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Key Concepts

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

Inequality Bounds
In mathematics, inequality bounds are used to describe a range of numbers that satisfy a certain condition. When you see an inequality like \(0 < x < 6\), this is telling you that the variable \(x\) is greater than 0 and less than 6. But here's a crucial detail: the numbers 0 and 6 are not part of the solution. They are bounds that define the limits within which \(x\) can lie. This type of inequality is known as a strict inequality because it does not allow \(x\) to equal the boundary values. Understanding inequality bounds helps in determining the set of solutions accurately and is the first step towards translating that into a graphical or numerical representation.
Graphing Intervals
Graphing intervals provides a visual representation of the range of possible values described by the inequality. When you have an interval such as \((0, 6)\), this can be graphically represented on a number line.
  • Plot an open circle at the number 0. This indicates 0 is not included in the interval.
  • Similarly, plot another open circle at the number 6, showing that 6 is also not included in the interval.
  • Draw a line connecting the open circles, which signifies all the values between 0 and 6 belong in the solution set.
Using open circles is crucial in correctly representing the inequality bounds, as it visually communicates that the endpoint values themselves are not part of the interval. Graphing intervals is particularly useful in recognizing the solution set's boundaries and confirming the interpretation of inequality in a tangible form.
Number Line Representation
The number line is a simple yet powerful tool used to visually display numbers and their relationships. For inequalities like \(0 < x < 6\), a number line helps illustrate the range of values \(x\) can take. The number line can encode information quickly and intuitively through the use of symbols and line segments.
  • First, start by drawing a horizontal line as your baseline.
  • Next, mark evenly spaced points that represent numbers; in this case, focus on the points 0 and 6.
  • The open circles placed on 0 and 6 are key in understanding that these endpoints are excluded.
  • The segment between them visually represents all the numbers that \(x\) can be from 0 to 6.
Number line representation is a hands-on way to depict numeric ranges and is extremely helpful in making abstract concepts from algebra more concrete and accessible.

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