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Chapter 2: Problem 103

Graph each inequality, and write it using interval notation. \(4>x>-3\)

Short Answer

Expert verified
The interval notation for \(4 > x > -3\) is \((-3, 4)\).

Step by step solution

01

Identify the inequality

The given inequality is \(4 > x > -3\). This means that the values of \(x\) are greater than \(-3\) and less than \(4\).
02

Rewrite the compound inequality

Rewrite the compound inequality as two separate inequalities: \(x > -3\) and \(x < 4\).
03

Understand the interval notation

The interval notation for values of \(x\) that are greater than \(-3\) and less than \(4\) is \((-3, 4)\). The parentheses indicate that the endpoints are not included in the interval.
04

Graph the inequality

On a number line, place an open circle at \(-3\) and at \(4\) to show that these points are not included. Shade the region between \(-3\) and \(4\) to represent all the values \(x\) can take.
05

Combine the information

The inequality \(4 > x > -3\) is represented on the number line by shading the region between \(-3\) and \(4\) with open circles at \(-3\) and \(4\). Using interval notation, the solution is \((-3, 4)\).

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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 method of writing subsets of the real number line. This method uses parentheses and brackets to describe intervals. Parentheses \( ( ) \) are used to denote that the endpoints are not included, while brackets \( [ ] \) include the endpoints.
In this exercise, the inequality \(4 > x > -3\) is rewritten in interval notation as \((-3, 4)\). The parentheses tell us that -3 and 4 are not part of the solution set, but all numbers between -3 and 4 are.
Compound Inequalities
Compound inequalities involve two separate inequalities that are combined into one statement by the word 'and' or 'or.' In this exercise, \(4 > x > -3\) is a compound inequality where two conditions must be true at the same time: \(x > -3\) and \ (x < 4\).
To handle a compound inequality:
  • Separate it into two parts: \ (x > -3) \ and \(x < 4) \.
  • Solve each part individually, but remember, both conditions apply simultaneously.
This means any value of \(x\) that we select must satisfy both \(x > -3\) and \(x < 4\).
Number Line Representation
A number line is a visual representation of numbers in order. It helps to show all possible solutions to an inequality.
For the inequality \(4 > x > -3\), we use a number line to visually represent the set of numbers that satisfy the inequality by using open circles and shading:
  • An open circle is placed at -3 and another at 4. This denotes that \-3 and 4 are not included in the solution.
  • The region between these circles is shaded to indicate that all numbers between -3 and 4 are included in the solution set.
When you see the shaded part on the number line between -3 and 4, it makes it clear which numbers satisfy the inequality. The open circles help distinguish that the endpoints are excluded in this case.

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Most popular questions from this chapter

Solve each equation for \(y\) $$ 4 x-y=1 $$

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