91Ó°ÊÓ

Algebra is an essential branch of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to create equations and inequalities, which can represent real-world situations. In the given exercise, we focus on an inequality involving a single variable, \(x\). Inequalities are like equations, but instead of showing equality, they show a relationship where one side is greater than or less than the other.

Algebra is a powerful tool for solving problems that would be too complicated to visualize or handle using basic arithmetic alone. When solving inequalities, we often need to perform various algebraic operations to find the solution set. This includes operations such as addition, subtraction, multiplication, and division. The key is to understand and apply the properties of inequalities correctly, which we will explore in the following sections.

When we multiply both sides of an inequality by the same number, it’s important to know whether the number is positive or negative. In inequalities, this can affect the direction of the inequality sign.

For example, if you multiply or divide both sides of an inequality by a positive number, the inequality sign remains the same. This is what we did in the exercise when multiplying both sides by 7 (a positive number), to isolate \(x\). Hence, the inequality direction was retained:

\[\frac{x}{7} \cdot 7 > -2 \cdot 7\]

Multiplying gave us \(x > -14\), which means \(x\) should be any value greater than -14.

If you multiply or divide by a negative number, the inequality sign flips. For instance, if we had divided by -7, we would need to reverse the inequality sign. Always remember this key rule:

• Positive multiplication: Direction stays the same.
• Negative multiplication: Direction reverses.

Positive numbers play a crucial role in maintaining the integrity of inequalities when performing operations. As seen in our step-by-step solution, dealing with inequalities can become complex without knowledge about how positive numbers affect the process.

When you multiply or divide an inequality by a positive number, the logical relationship between the expressions remains unchanged. This is an essential aspect of solving inequalities accurately.

Consider the following points when dealing with positive numbers in inequalities:

• The comparison's direction doesn't change when multiplying or dividing by positive numbers. This can make solving such inequalities straightforward and predictable.
• Understanding the properties of operations with positive numbers ensures inequality solutions are correct and consistent. For example, our exercise result was that \(x > -14\), indicating any number greater than -14 is a solution. Knowing this keeps us confidently solving similar problems.

By carefully applying these principles with positive numbers, solving inequalities becomes a logical and understandable task.