Problem 20 - Elementary Algebra Within Reach

Question

Match the statement with the property it represents. (a) Addition Property of Inequality (b) Subtraction Property of Inequality (c) Multiplication Property of Inequality (d) Division Property of Inequality \(-8>-9\), so \(-8+4>-9+4\).

Step-by-Step Solution

Answer

The statement represents the Addition Property of Inequality.

Step 1: Identify the Mathematical Operation

In the given equations, \(-8>-9\) and \(-8+4>-9+4\), the operation used is addition, rather than subtraction, multiplication, or division.

Step 2: Match the Operation with the Property

Since the operation used is addition, the property represented by these equations should be the Addition Property of Inequality. This property states that if a>b, then a+c>b+c for any real number c.

Key Concepts

Addition Property of Inequality

Let's dive into the Addition Property of Inequality, a fundamental concept when working with inequalities. This property essentially states that if you have an inequality such as \(a > b\), you can add the same number to both sides of the inequality without changing the inequality's direction. This is applicable for any real number \(c\). For instance, if \(-8 > -9\), by adding \(4\) to both sides, you get \(-8 + 4 > -9 + 4\), which simplifies to \(-4 > -5\). This confirms that the relationship between the two numbers remains greater than.

Understanding the Addition Property is crucial because it allows for modifications and solving of inequalities just like equations, but with an extra rule about not flipping the inequality sign as long as you are only adding terms.

Subtraction Property of Inequality

Shifting our focus to the Subtraction Property of Inequality, this rule allows you to subtract the same value from both sides of an inequality while maintaining its direction. It works similarly to the Addition Property but involves subtraction, as the name suggests.

This property can be represented mathematically as follows: if \(a > b\), then \(a - c > b - c\) for any real number \(c\). This consistency is vital in maintaining the integrity of the inequality throughout various mathematical manipulations.

When solving inequalities, remember the subtraction doesn't change the inequality's direction, providing opportunities to isolate variables or simplify terms effectively.

Multiplication Property of Inequality

The Multiplication Property of Inequality introduces an interesting twist. While multiplying both sides of an inequality by a positive number leaves the inequality unchanged, multiplying by a negative number reverses its direction.

The rule is as follows:

• If \(a > b\) and \(c > 0\), then \(ac > bc\).
• If \(a > b\) and \(c < 0\), then \(ac < bc\).

This distinction is crucial for accurate computation, emphasizing the importance of keeping track of the number's sign during multiplication.

Whenever engaging with inequalities, keep this property in mind, especially when solving for unknown variables as it ensures your solution preserves accuracy.

Division Property of Inequality

Finally, consider the Division Property of Inequality. This property operates similarly to multiplication but involves division of both sides of an inequality.

• Dividing both sides by a positive number retains the inequality's direction.
• Dividing both sides by a negative number flips the inequality's direction.

If you have \(a > b\) and \(c > 0\), then \(\frac{a}{c} > \frac{b}{c}\). Conversely, if \(c < 0\), \(\frac{a}{c} < \frac{b}{c}\).

This property is vital when dealing with equations and inequalities that require dividing by variables or constants, particularly when simplifying or solving for a variable while maintaining the relationship expressed by the inequality. Always account for the sign of the coefficient to ensure your conclusions are valid.