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Chapter 3: Problem 5

Write a verbal description of the inequality and sketch its graph. $$y<-9$$

Short Answer

Expert verified
The inequality \(y < -9\) represents all the values of \(y\) that are less than -9. In the graph, this would be represented by a dashed horizontal line at \(y = -9\) and shading below the line.

Step by step solution

01

Understanding the inequality

The inequality \(y < -9\) means that the value of \(y\) can be any real number less than -9.
02

Verbal description of the inequality

In words, the inequality would be described as: 'All the values of \(y\) that are less than -9'.
03

Sketching the graph

To graph this inequality, a horizontal line should be drawn at \(y = -9\). However, since \(y\) is less than -9 (not equal to), the line is going to be a dashed (or dotted) line, indicating that the points on the line aren't included. The shading (to represent all values less than -9) should be below this line.

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Key Concepts

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

Inequality Notation
Inequality notation is a mathematical way to express the relationships between two values or expressions. It tells us if one value is greater than, less than, greater than or equal to, or less than or equal to another value. In our example, the inequality is written as \(y < -9\). This notation indicates that \(y\) can take any value that is less than -9. When you see the symbol '<', it means 'less than'. Similarly, the symbol \('>'\) means 'greater than'. These symbols help to quickly convey what numbers can satisfy the inequality.
Verbal Description of Inequalities
When creating a verbal description of an inequality, you are translating the mathematical notation into words. This helps to visualize and understand the concept better. For the inequality \(y < -9\), we simply say, "all values of \(y\) are less than -9." This verbal description can help you better grasp what the inequality is communicating and can be useful when explaining it to someone else. It describes the set of all possible numbers that \(y\) can be without going into specific numbers.
Graphing Techniques
Graphing inequalities involves depicting them on a coordinate plane, which helps visualize the range of solutions. To graph \(y < -9\), you begin with a horizontal line at \(y = -9\). However, since \(y\) is strictly less than -9 and doesn't include -9 itself, a dashed line is used instead of a solid one. This indicates that the value where the line is drawn is not included in the set of solutions. Below this dashed line, you will shade the region, as this shading represents all the values that satisfy the inequality.
Graphing helps not only to understand inequalities but also to solve systems of inequalities by showing overlaps of solutions.
Real Numbers in Inequalities
When dealing with inequalities like \(y < -9\), we refer to real numbers. Real numbers include all the numbers on the number line, encompassing both rational numbers (like integers and fractions) and irrational numbers (like square roots and pi). By understanding real numbers, you grasp the idea that \(y\) in the inequality \(y < -9\) can be any decimal or integer value such as -10, -9.5, or even -1000.
This range of potential values is what makes inequalities dynamic. Real numbers ensure that inequalities are not restricted to just whole numbers or fractions, allowing for a complete representation of possible values that make an inequality true.

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Most popular questions from this chapter

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