Problem 82 - Algebra A Combined Function

Question

Fill in the box with \(<,>, \leq,\) or \(\geq\). See the Concept Check in this section. If \(m \leq n,\) then \(2 m \square 2 n\)

Step-by-Step Solution

Answer

The symbol is \( \leq \).

Step 1: Understand the Given Inequality

We are given that \( m \leq n \). This means that \( m \) is either less than or equal to \( n \). We need to determine the relationship between \( 2m \) and \( 2n \) based on this.

Step 2: Apply Multiplication to Both Sides

To find the relationship between \( 2m \) and \( 2n \), we multiply both sides of the inequality \( m \leq n \) by 2. This gives us \( 2m \leq 2n \).

Step 3: Choose the Correct Symbol

Since multiplying both sides of an inequality by a positive number (2, in this case) maintains the direction of the inequality, the inequality \( 2m \leq 2n \) stays as \( \leq \).

Key Concepts

Multiplication Property of Inequality

The multiplication property of inequality is a crucial concept in algebra that helps you understand how inequalities behave when you multiply both sides by a number. When the number is positive, like in our exercise, you can multiply both sides of the inequality without changing the inequality's direction. For example:

• If you start with an inequality like \( m \leq n \), multiplying each side by 2 will still keep it as \( 2m \leq 2n \).
• This happens because multiplying by positive numbers does not affect the order.

However, if you multiply each side by a negative number, the inequality sign must be reversed to maintain a true statement. For instance:

• If \( m > n \), and you multiply by \(-2\), it turns into \(-2m < -2n\).

Knowing these rules helps you solve inequality problems accurately. It shows how sensitive inequalities are to the operations we perform on them, especially multiplication and division.

Inequality Symbols

Inequality symbols allow us to express how values relate to each other, giving us a way to describe comparisons instead of just equivalences. Here's what each symbol means:

• \(<\): Less than - for example, \(m < n\) means \(m\) is smaller than \(n\).
• \(>\): Greater than - it shows that one quantity is larger than another.
• \(\leq\): Less than or equal to - this combines the \(<\) and the equality \(=\) symbols, indicating that a value can either be smaller or exactly equal.
• \(\geq\): Greater than or equal to - similarly, this symbol denotes that one value can be larger or exactly the same as the other.

Using these symbols correctly allows us to create precise mathematical statements about relationships between numbers and variables. It enables the solving and manipulation of mathematical expressions involving inequalities.

Mathematical Reasoning

Mathematical reasoning is the logical foundation needed to solve problems, like inequalities, effectively. It involves both inductive and deductive reasoning methods. Here's how it might apply to solving inequalities:

• Start with the premise. For instance, you know \( m \leq n \). This is your starting point and must be true for any following operations.


• Use logical steps and correct mathematical operations to manipulate the inequality to solve or further explore it. In our example, multiplying by 2 was a logical step.


• Finally, verify that your final statement or conclusion logically follows from the initial premise. When you confirm \( 2m \leq 2n \), you ensure that no step has altered the inequality incorrectly.

Good mathematical reasoning helps you maintain the integrity of solutions and ensures that each step leads to a valid conclusion. This approach is beneficial when assessing and solving complex mathematical situations.