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Chapter 1: Problem 35

State whether each inequality is true or false. (a) \(-3<-4\) (b) \(3<4\)

Short Answer

Expert verified
(a) False, (b) True.

Step by step solution

01

Analyze inequality (a)

In the inequality \(-3 < -4\), compare the two negative numbers on a number line. Remember that numbers further to the right are greater. Since \(-3\) is to the right of \(-4\) on a number line, conclude that \(-3\) is actually greater than \(-4\). Thus, the statement \(-3 < -4\) is false.
02

Analyze inequality (b)

In the inequality \(3 < 4\), compare the two positive numbers on a number line. Since 3 is to the left of 4, it means that 3 is indeed less than 4. Therefore, the statement \(3 < 4\) is true.

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Key Concepts

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

Number Line
A number line is a powerful tool we use to visually understand numeric relationships, including inequalities.
It displays numbers on a straight line, with equal intervals between them.
Positive numbers are typically to the right of zero, while negative numbers fall to the left.
Using a number line, you can easily compare the size of different numbers:
  • The further right a number is, the larger it is.
  • Conversely, the further left on the line, the smaller the number.
This feature allows us to quickly evaluate inequalities.
For example, when you place -3 and -4 on this line, you'll see -3 lies to the right of -4.
This indicates that -3 is greater, helping to check if an inequality statement is true or false.
Negative Numbers
Negative numbers are numbers that are less than zero.
They are used to represent values below zero and are found to the left side on a number line.
Key points about negative numbers include:
  • Every negative number is less than any positive number.
  • A larger negative number (in absolute value) implies a smaller value.
This means -3 is greater than -4 because it is less negative.
Think of a debt: If you owe $3, it is less severe than owing $4, hence closer to zero.
Understanding the position of negative numbers on a number line helps in determining inequalities.
Always remember: A lesser negative number is more significant in value compared to a greater negative one when comparing them.
Positive Numbers
Positive numbers are those greater than zero.
They appear to the right side of zero on a number line.
Generally, the following characteristics are true for positive numbers:
  • They are always greater than zero and any negative number.
  • Larger positive numbers are to the right of smaller positive numbers.
In comparing the example of 3 and 4, we notice 3 is to the left of 4 on the number line.
This confirms that 3 is indeed smaller than 4, adhering to their position on the line.
Handling positive numbers involves understanding they increase in value as you move to the right.
We use positive numbers extensively in everyday contexts, from measuring distance to counting objects.

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