91Ó°ÊÓ

Let's delve into the distributive property, a basic yet vital tool in algebra. This property allows us to remove parentheses in expressions, making complex equations more manageable. Here’s how it works. If you have an expression like \(a(b + c)\), you distribute \(a\) across the terms inside the parentheses.
This results in \(ab + ac\).
In our problem, we started with \(-4(x + 2)\) on one side of the inequality. By applying the distributive property, we multiplied \(-4\) by \(x\) and by \(+2\).
This step transformed the expression into \(-4x - 8\).
Using this property helps to simplify problems, setting the stage for further operations.

Solving inequalities isn't much different from solving equations, but there are key differences to keep in mind. Inequalities express a relationship showing that one side is larger or smaller than another, such as greater than (>), less than (<), or their inclusive versions (≥ or ≤).
After distributing and simplifying the terms in the given inequality, \(-4x - 8 > 3x + 20\), we began solving by consolidating like terms.
This involved moving variables to one side and constants to the other, ultimately leading to \(-7x > 28\).
The next crucial step involved solving these inequalities while preserving the inequality sign rules, vital in keeping the inequality true.

Isolating the variable is the fundamental goal in solving equations and inequalities, aiming to find out what value the variable represents. After consolidating terms with \(-7x > 28\), our next move was focused on isolating \(x\).
To do this, divide both sides by the coefficient of \(x\), which in this case is -7. However, there's a critical rule in inequalities.
Whenever you multiply or divide by a negative number, you must flip the inequality sign. This step transforms \(-7x > 28\) into \(x < -4\).
Understanding this rule is crucial to ensure the inequality represents its true mathematical relationship. Thus, through this methodical approach, we successfully isolated \(x\) and solved the inequality.