Problem 43
Question
For the following problems, what numbers can replace \(x\) so that the following statements are true? \(-3 \leq x \leq 2, \quad x\) a natural number
Step-by-Step Solution
Verified Answer
Answer: The natural numbers that can replace \(x\) are 1 and 2.
1Step 1: Identify the range of numbers
The inequality \(-3 \leq x \leq 2\) represents all numbers between -3 and 2, inclusive.
2Step 2: Find the natural numbers in the range
Natural numbers are positive integers greater than zero. In the range of \(-3 \leq x \leq 2\), the natural numbers are 1 and 2. Remember that 0 is not included in the set of natural numbers.
3Step 3: Write the final answer
The numbers that can replace \(x\) such that the inequality \(-3 \leq x \leq 2\) is true are the natural numbers 1 and 2.
Key Concepts
Inequality SolvingNatural NumbersNumber Range
Inequality Solving
Inequality solving involves finding the values that satisfy the given inequalities, which define the range of possible solutions. It's essential to follow a clear step-by-step approach to ensure no possible solutions are omitted.
Let's consider the inequality \( -3 \leq x \leq 2 \) with the additional condition that \( x \) must be a natural number. To solve this, one must first identify the entire set of numbers that \( x \) could represent, which includes all integers from -3 to 2. After this, we filter out those integers that are not natural numbers. Since natural numbers are all positive integers starting from 1, in this specific case, the solutions are limited to the natural numbers within the identified range. This methodical way of solving inequalities ensures that all solutions are considered and that none are accidentally overlooked.
Let's consider the inequality \( -3 \leq x \leq 2 \) with the additional condition that \( x \) must be a natural number. To solve this, one must first identify the entire set of numbers that \( x \) could represent, which includes all integers from -3 to 2. After this, we filter out those integers that are not natural numbers. Since natural numbers are all positive integers starting from 1, in this specific case, the solutions are limited to the natural numbers within the identified range. This methodical way of solving inequalities ensures that all solutions are considered and that none are accidentally overlooked.
Natural Numbers
Natural numbers are the basic building blocks in the world of mathematics, comprising all positive integers starting from 1. These are the numbers we use for counting and ordering objects in our day-to-day lives.
The concept of natural numbers excludes zero (0) and any negative numbers. It's important to remember this, especially when solving inequalities that involve natural numbers. Often, students confuse natural numbers with whole numbers, which include zero. However, for the purpose of solving our given inequality, only positive integers that are greater than zero will qualify as solutions.
The concept of natural numbers excludes zero (0) and any negative numbers. It's important to remember this, especially when solving inequalities that involve natural numbers. Often, students confuse natural numbers with whole numbers, which include zero. However, for the purpose of solving our given inequality, only positive integers that are greater than zero will qualify as solutions.
Number Range
A number range specifies the interval between two endpoints, where numbers can lie. The range can include all types of numbers—whole numbers, integers, or even decimals—depending on the context.
When working with inequalities, establishing the number range helps us visualize the potential solutions. In our exercise, the range is initially given by \( -3 \leq x \leq 2 \), which includes both negative and positive integers, as well as zero. As discussed earlier, to meet the natural number criteria, the range is then refined to exclude any non-positives, leaving us with a solution set range of only 1 and 2, which are the natural numbers falling within the original range.
When working with inequalities, establishing the number range helps us visualize the potential solutions. In our exercise, the range is initially given by \( -3 \leq x \leq 2 \), which includes both negative and positive integers, as well as zero. As discussed earlier, to meet the natural number criteria, the range is then refined to exclude any non-positives, leaving us with a solution set range of only 1 and 2, which are the natural numbers falling within the original range.
Other exercises in this chapter
Problem 43
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