Question

For what positive integers \(x\) is the statement \((x-2)^{2}+1 \leq 2\) true? For what integers is it true? For what real numbers is it true? If you expand the universe for which you are considering a statement about a variable, does this always increase the size of the statement's truth set?

Short answer

The solution is \(x = 1, 2, 3\) for positive integers, integers, and real numbers. Increasing the universe does not always increase the truth set size.

Step-by-step solution

Understand the Inequality

The given inequality is \((x-2)^{2}+1 \leq 2\). This means we need to find the set of values for \(x\) where this inequality holds true for different types of numbers: positive integers, integers, and real numbers.

Simplify the Inequality

Start by isolating the squared term. Subtract 1 from both sides of the inequality: \((x-2)^2 \leq 1\). This simplification will make it easier to determine the possible values of \(x\).

Solve the Inequality for Real Numbers

Consider the square root of both sides of \((x-2)^2 \leq 1\) to find the range of \(x\). For squared terms, this can be rewritten as two inequalities: \[-1 \leq x-2 \leq 1\].Solving these gives:1. \(-1 \leq x-2\) leads to \(1 \leq x\)2. \(x-2 \leq 1\) leads to \(x \leq 3\). Combining these, the solution for real \(x\) is \(1 \leq x \leq 3\).

Determine the Solution for Integers

Within the range found in Step 3 (\(1 \leq x \leq 3\)), identify the integer values. The integers that satisfy the inequality are \(x = 1, 2, 3\).

Determine the Solution for Positive Integers

From the integers found (\(x = 1, 2, 3\)), we identify the positive integers. All these numbers are positive, so the solution set for positive integers is also \(x = 1, 2, 3\).

Consider the Limitation Question

When we expand the universe from real numbers to just positive integers, we narrow the range of acceptable solutions, since not all real solutions are positive integers. Conversely, considering real numbers generally makes the set larger.

Key concepts

Positive Integers

Positive integers are the set of all counting numbers starting from 1 and going up indefinitely. These numbers do not include zero or any negative numbers. Positive integers involve

• Natural numbers that begin from 1.
• Whole numbers that do not have fractions or decimals.
• Numbers with a positive sign before them (or no sign, which implies positivity by default).

In the context of our inequality \((x-2)^2 + 1 \leq 2\), we are asked to find which positive integers meet this condition. By simplifying\((x-2)^2 \leq 1\), we find our range of valid values for positive integers is \(x = 1, 2, 3\). Recognizing positive integers helps in identifying solutions within a more narrow scope, which can be easier to comprehend than working with broader number types.

Integers

Integers encompass a broader category than positive integers. This set includes:

• All positive numbers from 1 upwards.
• Zero.
• All negative numbers going downwards without end.

They are whole numbers and do not include fractions or decimals. For the inequality \((x-2)^2 \leq 1\), the integer solution found by solving \[1 \leq x \leq 3\] includes \(x = 1, 2, 3\). Notice how this solution set includes all positive integers that satisfy the condition, plus any possible negative integers or zero where applicable. Here, the solution overlaps entirely with positive integers, which means more variations for different types of parameters in other cases would use the complete integer set.

Real Numbers

Real numbers are the most comprehensive collection of numbers we typically use in mathematics. This category comprises:

• Natural numbers and whole numbers.
• Decimal and fractional numbers.
• Both positive and negative numbers, and zero.
• Special numbers like irrational numbers (e.g., \(\pi\) and \(\sqrt{2}\)).

To solve for real numbers with our inequality \((x-2)^2 + 1 \leq 2\), simplifying gives us the range \[1 \leq x \leq 3\]. Thus, the solution for real numbers includes any number within and including the bounds of 1 and 3, whether it's a fraction, decimal, positive, negative, or zero. Real numbers provide the most complete picture, covering all possible intervals within the solution scope.

Truth Set

A truth set in mathematics refers to the set of all values that satisfy a particular condition or equation. It's like a club of values where membership depends on them making the inequality or equation true. In our case:

• For positive integers, the truth set is \(x = 1, 2, 3\).
• For integers, it remains \(x = 1, 2, 3\).
• For real numbers, the truth set is any number in the interval \([1, 3]\).

Changing the universe or expanding it affects the truth set because larger or different kinds of sets (like going from integers to real numbers) can either include more members, increasing its size, or restrict it due to added limitations.