Problem 17 - 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 \(10<12\), so \(\frac{10}{2}<\frac{12}{2}\).

Step-by-Step Solution

Answer

The statement '\(10<12\), so \(\frac{10}{2}<\frac{12}{2}\)' represents the Division Property of Inequality (d).

Step 1: Identify the operation

In the statement, \(10<12\), so \(\frac{10}{2}<\frac{12}{2}\), division is being applied to both sides of the inequality.

Step 2: Recognize the outcome

When each side of the inequality is divided by the same number (in this case, 2), the inequality remains valid.

Step 3: Match the term

The property of an inequality that allows division by the same non-zero number on both sides of the inequality without changing the inequality is known as the Division Property of Inequality.

Key Concepts

Properties of Inequality

Inequalities are mathematical statements that illustrate the relationship between two expressions that are not necessarily equal. Inequalities use symbols like less than (<) or greater than (>) to show this relationship. When we manipulate inequalities, there are certain rules and properties we need to remember. These properties of inequality ensure that the relationship between the expressions is preserved, even after performing operations on them.

Key properties include:

• Addition Property of Inequality: Adding the same number to both sides of an inequality does not change the direction of the inequality.
• Subtraction Property of Inequality: Subtracting the same number from both sides also keeps the inequality intact.
• Multiplication Property of Inequality: Multiplying both sides by the same positive number maintains the inequality. However, if you multiply both sides by a negative number, the direction of the inequality flips.
• Division Property of Inequality: Similar to multiplication, dividing both sides by the same positive number does not alter the inequality. If dividing by a negative number, the inequality reverses direction.

These properties are crucial for solving inequalities effectively and ensure consistency in mathematical reasoning.

Algebraic Inequalities

Algebraic inequalities are expressions that represent the relationship between variables and constants using inequality symbols like <, >, ≤, or ≥. Solving algebraic inequalities often involves similar steps to solving equations, but it's important to pay attention to the properties of inequality when manipulating them.

When solving algebraic inequalities, you often need to:

• Isolate the variable: This could involve using addition, subtraction, multiplication, or division to get the variable on one side of the inequality.
• Apply properties of inequality: This ensures the inequality remains valid as you work towards isolating the variable. Remember, multiplying or dividing by a negative number will reverse the inequality symbol.
• Check your solutions: It’s important to verify solutions by substituting them back into the original inequality to ensure they satisfy the inequality condition.

A good understanding of algebraic inequalities helps in tackling various real-world problems, such as calculating minimum or maximum values suitable for a given condition.

Mathematical Properties

Mathematical properties in inequality are rules that underline the consistent behavior of numbers and operations. These properties not only make calculations more manageable but also ensure that mathematical operations yield reliable results.

Key mathematical properties involved in inequalities include:

• Commutative Property: Pertains to addition and multiplication, stating that the order of numbers does not affect the result.
• Associative Property: Involves grouping of numbers in addition or multiplication without affecting the result.
• Distributive Property: Links addition and multiplication, illustrating that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding.

While these properties are general, their interaction with inequality properties is crucial. Understanding how these properties integrate with inequality concepts further enhances one's ability to solve complex mathematical problems effectively.