Problem 38 - Prealgebra

Question

Write an inequality that can be solved using the Division Property of Inequality, where the inequality symbol is not reversed.

Step-by-Step Solution

Answer

A suitable inequality is \(4x > 12\). When divided by 4, it becomes \(x > 3\), without reversing the inequality symbol.

Step 1: Understand the Division Property of Inequality

The Division Property of Inequality states that when you divide both sides of an inequality by a positive number, the direction of the inequality symbol does not change. Thus, the inequality remains the same.

Step 2: Choose a Simple Inequality

Let's use a simple inequality: \(4x > 12\). The goal is to isolate \(x\) using division.

Step 3: Divide Both Sides

Divide each side of the inequality \(4x > 12\) by 4, which is a positive number, so the direction of the inequality will not change. This will give us \(x > 3\).

Step 4: Verification

To verify our solution, substitute a number greater than 3 back into the original inequality to check if it holds true. For example, substitute \(x = 4\): \(4 \cdot 4 = 16\), which is greater than 12, hence the solution is correct.

Key Concepts

Inequality Solving

Solving inequalities involves finding the set of all possible values that make the inequality true. Compared to equations, where there is often one solution, inequalities can have a range of solutions. When solving linear inequalities, you'll manipulate the terms much like an equation to get the unknown (typically represented as a variable) by itself.

• For the inequality to remain true, you must treat both sides equally. If you add or subtract a number, do it to both sides.
• When it involves division or multiplication, especially through negative numbers, ensure you keep track of the inequality's direction.

The **Division Property of Inequality** is one such rule that applies during these operations, ensuring we retain the inequality's validity.

Mathematical Properties

Mathematical properties are universal truths or rules that help us solve problems consistently. In inequality solving, these properties give us tools to simplify problems. One of the essential properties in our context is the Division Property of Inequality, which applies when both sides of an inequality are divided by the same positive number.

• **Division Property of Inequality**: Dividing both sides of an inequality by a positive number does not change the inequality's direction.
• This property ensures balance and fairness in inequality expressions, maintaining their truthfulness.

Understanding such properties is crucial for solving mathematical problems efficiently and correctly. These properties allow us to transition appropriately through different steps of the solution process, like isolating variables or simplifying expressions.

Prealgebra Concepts

Prealgebra is the foundation of mathematics that prepares you for higher-level algebra topics. It includes understanding how to work with basic operations and inequalities. In prealgebra, you'll come across concepts such as:

• **Variables and Expressions**: Recognize and manipulate variables, which are placeholders for numbers.
• **Operation Understanding**: Understanding how operations like addition, subtraction, multiplication, and division impact numbers.
• **Inequalities**: Grasping the notion that inequalities represent a range of possible solutions as opposed to a single answer.

In the context of inequalities, prealgebra teaches us the rules and properties, such as the Division Property of Inequality, which aids in comprehending how these rules apply when isolating a variable. Building a robust understanding in these areas allows you to comfortably transition into more complex algebraic tasks.