Problem 83 - 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 \( \geq \).

Step 1: Understanding Inequality

We start with the inequality given: \( m \leq n \). This implies that \( m \) is either less than or equal to \( n \). We need to determine what the inequality becomes when both sides are multiplied by -2.

Step 2: Multiplying by Negative Number

When we multiply both sides of an inequality by a negative number, we must reverse the inequality sign. Thus, taking the inequality \( m \leq n \) and multiplying both sides by \(-2\), we have: \(-2m \geq -2n\).

Step 3: Filling in the Correct Symbol

The inequality \(-2m \geq -2n\) indicates that we need the symbol \( \geq \) to correctly fill in the box.

Key Concepts

Multiplying Inequalities

Understanding inequalities is essential when dealing with math problems involving two expressions. When we multiply or divide both sides of an inequality by a positive number, the inequality sign remains the same. However, multiplying both sides of an inequality by a negative number requires careful attention.

• Multiplying both sides of an inequality by the same positive number keeps the inequality sign the same.
• Multiplying both sides of an inequality by a negative number requires reversing the inequality sign.

For example, if we have an inequality such as \( m \leq n \) and we multiply each side by \(-2\), it changes to \(-2m \geq -2n\). The key here is to remember to reverse the inequality sign due to the negative multiplier.

Reversing Inequality Sign

Reversing the inequality sign might seem tricky, but it's straightforward once understood. The main rule is: whenever you multiply or divide each side of an inequality by a negative number, flip the inequality sign. This step ensures that the relationship between the two expressions remains accurate.

• When starting with \( a < b \), multiplying both sides by \(-1\) results in \(-a > -b\).
• The rule keeps the relative magnitude of the numbers but changes the order representation.

In problems like our exercise, reversing the inequality sign ensures that the logic of the inequality holds true under transformations that involve negatives.

Negative Numbers in Inequalities

Negative numbers in inequalities can often introduce a need for additional considerations, specifically related to reversing the inequality. Understanding how negative numbers interact in a computation involving an inequality is crucial. Here's what you should keep in mind:

• Adding or subtracting a negative number from both sides doesn't change the inequality direction.
• Multiplying or dividing by a negative number requires flipping the inequality.

For example, in our exercise, multiplying by \(-2\) required a reversal of the inequality, turning \( m \leq n \) into \(-2m \geq -2n\). When working with negatives, always consider the impact they have on the inequality sign.

Algebraic Expressions

Algebraic expressions consist of variables, coefficients, and constants combined through arithmetic operations. In inequalities, these expressions serve as the foundation of comparison.

• Terms in an algebraic expression can involve both positive and negative coefficients.
• Understanding how operations like addition, subtraction, multiplication, and division affect inequality is crucial.

For instance, in our inequality \( m \leq n \), both \( m \) and \( n \) can represent algebraic expressions. Manipulating such expressions, while keeping track of inequality rules, is a significant skill in algebra. When multiplied by \(-2\), it was critical to apply the rules correctly to maintain the inequality's integrity.