Problem 14 - Fundamentals of Mathematics

Question

What numbers can replace \(x\) so that each statement is true? \(-3

Step-by-Step Solution

Answer

The numbers 1 and 2 can replace \(x\).

Step 1: Understanding the Range

The given inequality for the number is \[-3 < x < 3.\] This means that any number that replaces \( x \) must be greater than \(-3\) and less than \(3\). However, this is just the first part of the condition.

Step 2: Definition of Natural Numbers

Recall that natural numbers are the set of positive integers starting from 1, i.e., \(\{1, 2, 3, 4, \ldots\}\) These numbers are positive and do not include zero or any negative numbers. Hence, \(x\) must be a positive integer to satisfy being a natural number.

Step 3: Applying Both Conditions

Now, combine the two conditions. Given the inequality \(-3 < x < 3\) and \(x\) being a natural number:- The only natural numbers greater than \(-3\) and less than \(3\) are 1 and 2.

Step 4: Conclusion

The numbers that can replace \(x\), satisfying both conditions, are 1 and 2.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols like \(<\), \(>\), \(\leq\), and \(\geq\) to indicate whether one number is less than, greater than, or equal in comparison to another. In our example, the inequality \(-3 < x < 3\) tells us that the value of \(x\) is "sandwiched" between -3 and 3. This means:

• \(x\) must be greater than -3,
• and at the same time, \(x\) must be less than 3.

These constraints limit the possible values \(x\) can have, providing a range of values that \(x\) could potentially take. Inequalities like these are very useful in both math and real-world situations where exact numbers cannot always be determined. They help in understanding limits and boundaries of values.

Positive Integers

Positive integers are all the whole numbers greater than 0. They form a well-known sequence starting from 1 and continue upwards without any end. No fractions, decimals, or negative numbers are included in positive integers.

• This set includes numbers like 1, 2, 3, and so forth,
• and does not include numbers like 0, -1, or 1.5.

Recognizing that natural numbers are, in fact, a subset of positive integers, helps grasp their overall behavior. Natural numbers start at 1 and count upwards, much like positive integers. So, when considering the values that satisfy both the inequality and being a natural number, they must:

• be positive, and
• be a whole number.

Number Sets

Number sets are groups of numbers that share a particular property or pattern. They help to organize numbers so that we can better understand and manipulate them in mathematical expressions and equations. Let's look at some common number sets:

• Natural numbers \(\{1, 2, 3, \ldots\}\) start at 1 and go upwards, used for counting.
• Whole numbers \(\{0, 1, 2, \ldots\}\) start at 0 and include all natural numbers.
• Integers \(\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}\) include whole numbers and their negative counterparts.
• Rational numbers include fractions or ratios of integers.

By using set notation and understanding the unique properties of each set, solving problems that involve multiple types of numbers or determining a particular number's set classification becomes straightforward. In this exercise, we focus on natural numbers as they pertain to our inequality, further narrowing the range of possible values for \(x\).