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Chapter 2: Problem 91

Write inequality in interval notation, and graph the interval. \(4>x>-3\)

Short Answer

Expert verified
The interval in interval notation is \((-3, 4)\).

Step by step solution

01

Rewrite the Inequality

The given inequality is written as a compound inequality: \[4 > x > -3\]This means that x is a value that is greater than -3 and less than 4.
02

Identify the Interval

Recognize that the compound inequality implies that x lies between -3 and 4. Both endpoints are not included, as the inequality signs are strict (greater than and less than, but not equal to).
03

Write in Interval Notation

In interval notation, an open interval is used when the endpoints are not included. Therefore, the interval for the inequality \(4 > x > -3\) will be written as: \((-3, 4)\)
04

Graph the Interval

To graph the interval, plot an open circle at -3 and an open circle at 4 on a number line. Shade the region between these two points to represent all the numbers between -3 and 4 (but not including -3 and 4).

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Key Concepts

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

Compound Inequality
A compound inequality involves two separate inequalities that are joined by the words 'and' or 'or'. When we combine two inequalities using 'and', we are looking for values that satisfy both conditions simultaneously. For example, the compound inequality \(4 > x > -3\) means that x has to be more than -3 'and' less than 4 at the same time. This indicates a range of possible values for x, where x is confined between -3 and 4, but does not include -3 and 4 themselves.
Open Interval
An open interval represents a range of numbers between two endpoints but does not include the endpoints themselves. In interval notation, open intervals are denoted with round brackets (parentheses). For example, the inequality \(4 > x > -3\) translates to the open interval \((-3, 4)\). This notation tells us that x can be any value greater than -3 and less than 4, but not equal to either endpoint.
Graphing Inequalities
Graphing inequalities on a number line helps to visually represent the solution set. For the inequality \(4 > x > -3\), follow these steps:
  • Draw a number line and mark the points -3 and 4.
  • Place an open circle at -3 and another open circle at 4. This indicates that these endpoints are not included in the set of solutions.
  • Shade the region between -3 and 4 to show that every number in this range is a solution to the inequality.
By doing this, anyone can easily see the range of values that satisfy the inequality.

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