Problem 25 - Algebra A Combined Function

Question

Rewrite each inequality so that the inequality symbol points in the opposite direction and the resulting statement has the same meaning as the given one. $$ -10>-12 $$

Step-by-Step Solution

Answer

The reversed inequality is \(-12 < -10\).

Step 1: Identify the Initial Inequality

The given inequality is \(-10 > -12\). We need to rewrite it such that the inequality symbol points in the opposite direction.

Step 2: Reverse the Inequality Symbol

Change the '>' symbol to '<' to reverse the direction of the inequality. This step will involve changing both sides of the inequality.

Step 3: Swap the Sides

To maintain the original meaning of the inequality, swap the sides of the inequality. This gives us \(-12 < -10\), which says the same thing as \(-10 > -12\).

Key Concepts

Inequality Symbols

Inequality symbols are crucial in mathematics to represent the relationship between two values. These symbols include:

• Greater than: \(>\)
• Less than: \( < \)
• Greater than or equal to: \( \geq \)
• Less than or equal to: \( \leq \)

These symbols help express concepts like who has more or what is smaller. For example, if we compare -10 and -12 using the inequality \(-10 > -12\), it effectively states that -10 is greater than -12, which makes sense on the number line. It might seem odd, since both are negative numbers, but remember that the number closer to zero is always greater.

Understanding inequality symbols is the first step to working efficiently with inequalities because they help us define what we are comparing and how. Recognizing these symbols allows for the interpretation of mathematical statements and ensures that we can correctly understand what each statement means.

Reverse Inequality

Reversing an inequality is a common mathematical maneuver required to maintain an expression while changing its perspective. When reversing the inequality in an expression, you turn the inequality symbol around. In our example, the inequality \(-10 > -12\) is reversed to \(-12 < -10\). The change from 'greater than' to 'less than' keeps the mathematical statement true, but shows it from another angle.

To reverse an inequality correctly:

• Change the inequality symbol: \(>\) becomes \( < \), and vice versa.
• Swap the sides of the values involved.

This swap does not alter the relative sizes of the values involved but it does present the statement in another form. For example, saying "-10 is greater than -12" is the same as saying "-12 is less than -10". Both statements are different ways of describing the same numerical relationship. The process keeps the inequality logically equivalent.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations or inequalities to find solutions or to restate an equation in a different form without changing its meaning. It's a skillful technique used to simplify problems and make them more understandable.

When dealing with inequalities like \(-10 > -12\), algebraic manipulation allows us to reverse the direction while maintaining the integrity of its statement. This manipulation requires:

• Swapping terms across the inequality symbol.
• Reversing the inequality symbol to preserve the logical meaning.

Such manipulations are not only limited to inequalities, but they form the basis of solving many algebraic equations. Practicing algebraic manipulation helps in understanding how equations and inequalities can be restructured. This enhances mathematical intuition, making it easier to solve complex problems logically and methodically. When you master these simple actions, more complex algebraic challenges become easier to tackle.