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Chapter 2: Problem 6

Classify each pair of inequalities as "equivalent" or "not equivalent. $$ t>-1 ;-1

Short Answer

Expert verified
The pair of inequalities are equivalent.

Step by step solution

01

Analyze the first inequality

The first inequality is given as: ewline ewline t > -1 ewline ewline Analyze what numbers satisfy this inequality. This means any number that is greater than -1.
02

Analyze the second inequality

The second inequality is given as: ewline ewline -1 < t ewline ewline Analyze what numbers satisfy this inequality. This means any number that is greater than -1.
03

Compare the inequalities

Compare the two inequalities ewline ewline t > -1 ewline ewline and ewline ewline -1 < t ewline ewline Notice that both inequalities describe the exact same set of numbers: ewline ewline Any number greater than -1.
04

Classify the pair of inequalities

Since both inequalities describe the same set of values, the pair of inequalities: ewline ewline t > -1 ewline ewline and ewline ewline -1 < t ewline ewline are equivalent.

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Key Concepts

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

Equivalent Inequalities
Inequalities can sometimes look different but describe the same set of numbers. These are called 'equivalent inequalities'. For instance, consider the inequalities:

1. \( t > -1 \) 2. \( -1 < t \)

Both describe the set of numbers greater than -1. Even though they are written differently, they mean the same thing. You can think of equivalent inequalities as saying the same thing in different ways. This concept is useful because it allows for flexibility in mathematical expressions.

When working with inequalities, always remember that flipping the inequality sign accompanies flipping the sides of the inequality.
Inequality Analysis
Analyzing inequalities is about understanding what numbers satisfy them. Let's break down the process using our example: \( t > -1 \). Here are steps to analyze it:
  • Identify the inequality sign (>).
  • Determine the boundary (-1 in this case).
  • Note that the inequality does not include -1 itself, only numbers greater than -1.
By identifying these parts, one can easily understand what values make the inequality true.

For the second inequality \( -1 < t \), the process is similar. Notice that the sign (<) tells us that \( t \) must be more than -1. By comparing the two, you can see they both describe numbers greater than -1. A clear understanding of these steps ensures you grasp what each inequality represents.
Comparison of Inequalities
When comparing inequalities, determine if they describe the same range of numbers. Given: \( t > -1 \) and \( -1 < t \), both explain that \( t \) must be greater than -1. Despite different appearances, they are equivalent since they encompass identical sets of numbers.

To compare different inequalities:
  • Write down what each inequality states.
  • Determine the values that satisfy each one.
  • Check if these values overlap completely.

If the ranges match, the inequalities are equivalent. Understanding this is fundamental in solving and simplifying math problems involving inequalities.

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