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Chapter 2: Problem 89

Write inequality in interval notation, and graph the interval. \(0< y \leq 10\)

Short Answer

Expert verified
(0, 10]

Step by step solution

01

- Understand the Inequality

Examine the given inequality: \(0 < y \leq 10\)This means that \(y\) is greater than 0 and less than or equal to 10.
02

- Express in Interval Notation

Translate the inequality into interval notation. Since \(0\) is not included (open interval) and \(10\) is included (closed interval), the interval notation is \((0, 10]\).
03

- Graph the Interval

To graph the interval \((0,10]\) on a number line:1. Draw a number line.2. Put an open circle at \(0\) to indicate \(0\) is not included in the interval.3. Put a closed circle or bracket at \(10\) to indicate \(10\) is included in the interval.4. Shade the region between \(0\) and \(10\).

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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 expressions using inequality signs like <, >, 鈮�, or 鈮�. Essentially, it tells us that one side of the inequality is either smaller or larger than the other or in some cases, possibly equal.

For example, the inequality \(0 < y 鈮� 10\) means that \(y\) has values greater than 0 but less than or equal to 10.

It鈥檚 important to identify the direction of the inequality and whether the end value is included (鈮� or 鈮�) or not included (< or >). This helps in correctly translating it to interval notation and drawing the right graph on the number line.
graphing intervals
Graphing intervals involves marking the range of values on a number line. Here, you visually represent the solution set of an inequality.

Consider the inequality \(0 < y 鈮� 10\). To graph this:
  • Draw a number line.
  • Since 0 is not included, place an open circle at 0.
  • Since 10 is included, place a closed circle or a bracket at 10.
  • Shade the region between 0 (not included) and 10 (included).
This visual helps better understand the solution set, showing exactly which values fit the inequality.
number line graph
A number line graph is a visual tool to represent numbers and intervals. This tool is particularly useful for displaying mathematical concepts like inequalities and intervals.

For the inequality \(0 < y 鈮� 10\):
  • First, draw a horizontal line representing the number line.
  • Mark points for 0 and 10 on this line.
  • Place an open circle at 0, signaling it鈥檚 not included in the interval.
  • Put a closed circle or bracket at 10 to show it's included.
  • Finally, shade the section of the line between 0 and 10 to illustrate the interval \((0, 10]\).
This graphical representation helps learners quickly identify the range of acceptable values for \(y\) without relying solely on text.

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