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Magazine Chapter 2: Problem 20
Graph each inequality and describe the graph using interval notation. $$ x \geq-2 $$
Short Answer
Expert verified
Graph as a solid dot on \(-2\) with a line to the right; interval: \([-2, \infty)\).
Step by step solution
01
Understand the Inequality
The inequality given is \(x \geq -2\). This means we need to find all values of \(x\) that are greater than or equal to \(-2\). The symbol \(\geq\) indicates that \(-2\) is included in the solution set.
02
Draw the Number Line
To graph the inequality \(x \geq -2\), start by drawing a horizontal number line. Mark the point \(-2\) on the number line.
03
Graph the Inequality
Because \(x\) can equal \(-2\), place a solid dot at \(-2\) to indicate it is included in the set of solutions. Then, draw a line or arrow extending to the right from \(-2\), indicating that all numbers greater than \(-2\) are also included.
04
Write in Interval Notation
Interval notation for this graph starts at \(-2\) and goes to positive infinity, because all numbers greater than or equal to \(-2\) are included. In interval notation, this is written as \([-2, \infty)\). The square bracket at \(-2\) indicates that \(-2\) is included, and the parenthesis at infinity indicates that it extends indefinitely.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a concise way to describe sets of numbers, particularly the range of solutions for inequalities. In the case of the inequality \(x \geq -2\), we express it using interval notation as \([-2, \infty)\). This expression consists of:
- The smallest value in the solution set, \(-2\), represented at the beginning with a square bracket \([\). This square bracket signifies that \(-2\) is included in the solution set.
- Infinity \(\infty\), which suggests that the numbers continue endlessly in a positive direction. In interval notation, infinity is always accompanied by a parenthesis \(()\) because infinity itself is not a number that can be reached or included.
Number Line
A number line is a visual representation of numbers laid out in a horizontal line. It is an essential tool in graphing inequalities. To graph \(x \geq -2\), follow these steps:
- Draw a straight horizontal line.
- Mark equal sections along the line for clarity. Include the point \(-2\), which will be significant in this problem.
Solution Set
The solution set of an inequality consists of all values of \(x\) that make the inequality true. For \(x \geq -2\), the solution set not only includes \(-2\) itself but also every number greater than \(-2\). On a number line, this is shown by:
- A solid dot on \(-2\), indicating that \(-2\) is part of the solution set.
- A continuous extension to the right with an arrow, demonstrating all values greater than \(-2\).
Solid Dot Representation
In graphing inequalities on a number line, specific symbols help describe different conditions. The solid dot is crucial because it uniquely indicates inclusion. When you see a solid dot on a number line, it means that specific value belongs to the solution set. For the inequality \(x \geq -2\):
- The solid dot at \(-2\) indicates that the number \(-2\) itself is part of the solution set. This correlates with the "greater than or equal to" part of the inequality.
- Without this dot, \(-2\) would not be included, changing the interpretation and the solution set significantly.
