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Chapter 4: Problem 42

Graph and write interval notation for each compound inequality. $$ t \geq 2 \text { or }-3>t $$

Short Answer

Expert verified
( -\infty, -3 ) \cup [ 2, \infty )

Step by step solution

01

Understand the Compound Inequality

Identify the two parts of the compound inequality: 1) \( t \, \geq \, 2 \) and 2) \( -3 \, > \, t \). The word 'or' means that values satisfying either one of these conditions are solutions to the inequality.
02

Graph Each Inequality

For \( t \, \geq \, 2 \), plot a solid dot on 2 and shade to the right. For \( -3 \, > \, t \), plot an open dot on -3 and shade to the left.
03

Combine the Graphs

Combine the graphs of the two inequalities to show the entire solution set. There will be shading to the left of -3 and to the right of 2. Ensure that these segments do not connect.
04

Write the Interval Notation

The interval notation for the combined solution is \( ( - \infty, -3 ) \cup [ 2, + \infty ) \). This captures both parts of the solution set.

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Key Concepts

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

Understanding Interval Notation
Interval notation is a way of representing a range of values on the number line. It uses parentheses \(( \) and \(( \) brackets to describe the endpoints of the interval.

There are a few key concepts to remember:
  • Parentheses \(( \) indicate that an endpoint is not included (open interval).
  • Brackets \([ \)] indicate that an endpoint is included (closed interval).
  • Intervals spanning to positive or negative infinity always use parentheses, as infinity is not a fixed, reachable number.
In the given problem, two inequalities are combined using the word 'or', meaning any value that satisfies at least one part of the inequality is part of the solution set.

The interval notation for the compound inequality \( t \, \geq \, 2 \text { or }-3>t \) is \( ( - \infty, -3 ) \cup [ 2, + \infty ) \). This means all numbers less than -3 and all numbers greater than or equal to 2.
Inequality Graphing
Graphing inequalities helps visualize the range of values satisfying the conditions. Each inequality has its own graph representation:
  • For \( t \, \geq \, 2 \), draw a solid dot on 2 to show that 2 is included in the solution. Then shade to the right to indicate all values greater than 2.
  • For \( -3 \, > \, t \), draw an open dot on -3 to show that -3 is not included in the solution. Shade to the left to indicate all values less than -3.
When graphing compound inequalities with 'or', you combine these individual graphs. This means you will have two separate shaded regions:
  • One from negative infinity to -3, not including -3.
  • Another from 2 to positive infinity, including 2.
Combining these graphs shows the complete solution to the compound inequality.
Solution Set Explanation
The solution set of a compound inequality includes all values that satisfy at least one part of the inequality. For the given inequality \( t \, \geq \, 2 \text { or }-3>t \), the solution set is:

  • All values less than -3 \(( - \infty, -3 )\).
  • All values greater than or equal to 2 \([ 2, + \infty )\).
This means that any number in these intervals satisfies one part of the compound inequality.

Writing the solution set in interval notation, we use the union symbol \( \cup \) which indicates the combination of the two sets:

\( ( - \infty, -3 ) \cup [ 2, + \infty ) \)

This notation concisely expresses all the possible values that solve the inequality. Understanding interval notation and how to graph inequality are fundamental skills in solving these types of problems efficiently.

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Most popular questions from this chapter

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