Problem 76 - Elementary Algebra

Question

What whole numbers can replace \(x\) so that the statement \(-2 \leq x<2\) is true?

Step-by-Step Solution

Answer

Answer: The whole numbers that can replace \(x\) for the given inequality are \(-2, -1, 0,\) and \(1\).

Step 1: Write down the inequality

The given inequality is \(-2 \leq x < 2\).

Step 2: List the whole numbers in the range of the inequality

The whole numbers in the range of the inequality are \(-2, -1, 0, 1\).

Step 3: Determine which numbers satisfy the inequality

Now, let's check each number to see if it satisfies the inequality: \begin{equation} \begin{cases} -2 \leq -2 < 2 & \text{True} \\ -2 \leq -1 < 2 & \text{True} \\ -2 \leq 0 < 2 & \text{True} \\ -2 \leq 1 < 2 & \text{True} \end{cases} \end{equation}

Step 4: Write the answer

All the numbers \(-2, -1, 0,\) and \(1\) satisfy the inequality \(-2 \leq x < 2\). So, these are the whole numbers that can replace \(x\).

Key Concepts

Whole Numbers

Whole numbers are a fundamental concept in mathematics. They include all the numbers starting from 0 and extend to positive numbers: 0, 1, 2, 3, and so on.
Whole numbers do not include negative numbers or fractions, which makes them easy to understand and work with.
In problems involving inequalities, identifying whole numbers is essential as they form part of the possible solutions.

• Natural Numbers: These are the numbers used for counting (1, 2, 3, ...). Natural numbers are a subset of whole numbers.
• Integers: Integers include all whole numbers and their negative counterparts (-1, -2, ...), as well as zero.

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions.
It is used to generalize operations and relationships between numbers. Algebraic expressions are especially useful for solving problems with unknown values.
In the context of inequalities, algebra helps in representing relationships and solving for unknown variables using equations and inequalities.

• Variables: Symbols like \(x\), \(y\) are used to represent numbers. They allow us to express equations like \(-2 \leq x < 2\).
• Equations vs. Inequalities: While equations like \(2x = 6\) show equality, inequalities like \(-2 \leq x < 2\) express a range of values that the variable can take.

Algebra is essential for understanding how inequalities work and finding solutions that satisfy them.

Range of Inequality

The range in an inequality refers to all the values that a variable can take to satisfy the inequality. In the inequality \(-2 \leq x < 2\), the range comprises all numbers that make the statement true.
Determining this range involves finding all numbers that lie between the given bounds, including one bound but not the other in some cases.

• End Points: In \(-2 \leq x < 2\), \(-2\) is included in the range (as indicated by "\(\leq\)") while \(2\) is not (indicated by "\(<\)").
• Testing Values: Checking the whole numbers between the boundaries confirms the range. If each number satisfies the inequality's condition, it is part of the solution set.

Understanding the range helps ensure that all possible solutions are considered, ensuring a comprehensive approach to solving inequalities.