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Chapter 9: Problem 56

For each compound inequality, decide whether intersection or union should be used. Then give the solution set in both interval and graph form. $$ x>-1 \text { and } x<7 $$

Short Answer

Expert verified
Intersection. Interval: \((-1, 7)\).

Step by step solution

01

- Identify the type of compound inequality

The compound inequality given is: \ \( x > -1 \) \ and \ \( x < 7 \). \ Since the inequalities are connected by 'and', this means both conditions must be true simultaneously. Therefore, the solution is an intersection of the two inequalities.
02

- Solve each individual inequality

Solve the individual inequalities separately: \ 1. \( x > -1 \) means all values greater than -1. \ 2. \( x < 7 \) means all values less than 7.
03

- Find the intersection of the two solutions

The intersection of \( x > -1 \) and \( x < 7 \) is the set of values that satisfy both conditions. This is the interval: \ \( -1 < x < 7 \).
04

- Express the solution set in interval notation

In interval notation, the solution set is written as: \ \( (-1, 7) \).
05

- Graph the solution set

On a number line, the solution set \( -1 < x < 7 \) can be graphed by drawing a line between -1 and 7 with open circles at -1 and 7 to indicate that these endpoints are not 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.

Intersection of Inequalities
In compound inequalities, 'and' indicates an intersection of inequalities. This means both conditions must be true simultaneously. For example, given the inequalities\( x > -1 \) and\( x < 7 \), the solution is the overlap where both conditions are met. If you visualize each condition on a number line:
  • \( x > -1 \) means all values to the right of -1.
  • \( x < 7 \) means all values to the left of 7.
The overlap, or intersection, occurs between -1 and 7. So,\( -1 < x < 7 \)represents the values that satisfy both inequalities simultaneously.
Interval Notation
Interval notation is a way of writing subsets of the real number line. It uses brackets and parentheses to describe an interval. For example:
  • \( (a, b) \) represents all numbers between a and b, but not including a and b.
  • \( [a, b] \) includes both a and b.
For the inequality\( -1 < x < 7 \), we use parentheses because we do not include the end values -1 and 7. Therefore, in interval notation, the solution is\( (-1, 7) \). This notation helps to quickly identify the range of the solution.
Graph Solutions
Graphing solutions of inequalities on a number line helps visualize the range of values that satisfy the inequality. For the solution \( -1 < x < 7 \):
  • Draw a line from -1 to 7.
  • Use open circles at -1 and 7 to indicate these values are not included in the set.
This visual representation helps in understanding the solution's extent quickly and effectively. Graphing is a powerful tool in learning and verifying solutions to inequalities.

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Most popular questions from this chapter

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