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Chapter 0: Problem 6

Rewrite in interval notation and graph. $$x > -3 \text { and } x \leq 2$$

Short Answer

Expert verified
Interval notation: \((-3, 2]\). Graph: open at \(-3\), closed at 2.

Step by step solution

01

Determine the Inequalities

We are given two inequalities: \( x > -3 \) and \( x \leq 2 \). Our goal is to express the solution of these inequalities in interval notation.
02

Understand the Intersection

Both conditions must be simultaneously true. Thus, we require the intersection of the sets \( \{ x \mid x > -3 \} \) and \( \{ x \mid x \leq 2 \} \).
03

Express in Interval Notation

The solution set, where both conditions \( x > -3 \) and \( x \leq 2 \) are satisfied, is all numbers between \(-3\) and \(2\). In interval notation, this is written as \((-3, 2]\).
04

Graph the Interval

On the number line, draw a line starting just after \(-3\) up to \(2\). At \(-3\), draw an open circle because \(-3\) is not included in the solution set. At 2, draw a closed dot because \(2\) is included in the solution set.

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Key Concepts

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

Understanding Inequalities
Inequalities are mathematical expressions used to compare two values. They help us describe a range of values that a variable can take. When you see a symbol like ">" or "\( \leq \)", these indicate an inequality.

For instance, in the expression \(x > -3\), we are saying that \(x\) can be any number greater than \(-3\). We do not include \(-3\) itself.
On the other hand, \(x \leq 2\) means that \(x\) can be any number that is less than or equal to 2, where 2 is included in the possible values.
  • "Greater than" and "less than" signs (>, <) mean the specific number is **not** included.
  • "Greater than or equal to" and "less than or equal to" signs (\( \geq, \leq \)) mean the specific number **is** included.
Understanding inequalities helps us solve problems where a variable can have a range of possible values.
Graphing on a Number Line
A number line graph visually represents the solution of inequalities. It helps us quickly see which parts of the number line satisfy the inequality.

To graph \((-3, 2]\) on a number line, we represent different parts of the interval with circles or dots:
  • Start by drawing an open circle at \(-3\). The open circle means that \(-3\) is not part of the solution.
  • Draw a closed dot at \(2\) to show that \(2\) is included in the solution set.
  • Then, draw a solid line connecting the open circle and closed dot, representing numbers greater than \(-3\) but less than or equal to \(2\). This solid line indicates all numbers in between are part of the solution.
Graphing inequalities helps provide a clear image of the set of numbers we're dealing with.
Finding the Solution Set
The solution set of an inequality includes all numbers that satisfy the condition given by the inequality.

For the inequalities \(x > -3\) and \(x \leq 2\), the solution set is the intersection of both conditions.
This means finding numbers where both inequalities are true at the same time.
  • The numbers satisfying \(x > -3\) are all numbers greater than \(-3\).
  • For \(x \leq 2\), it includes all numbers up to and including 2.
  • The intersection point of these two sets鈥攚here the conditions overlap鈥攊s from \(-3\) (not included) to \(2\) (included).
This combined solution is represented as \((-3, 2]\) in interval notation, indicating the complete set of numbers that satisfy both inequalities.

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