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The process of solving inequalities often requires transforming the given inequality into a simpler form. This transformation involves performing the same operation on both sides of the inequality, very much like you would with an equation. The main operations you can perform are:

• Addition or subtraction of the same number or expression on both sides.
• Multiplication or division by the same positive number on both sides.

Each of these operations helps isolate the variable, making it easier to find the solution. However, there are additional rules to consider when working with inequalities, especially with negative numbers. These are pivotal to ensuring the inequality remains true after transformation.

Dividing both sides of an inequality by a number is a common technique used to simplify the expressions and solve for the variable. When dividing by a positive number, the direction of the inequality remains unchanged. However, there’s a crucial rule that stands out here: when you divide both sides of an inequality by a negative number, you must reverse, or flip, the inequality sign.
For example, if you start with \[-10a > 100\], you need to divide both sides by -10 to isolate \(a\). Since you are dividing by a negative number, the inequality sign flips from \(>\) to \(<\), resulting in \(a < -10\). This flipping is essential because multiplying or dividing by a negative alters the order of the values.

Negative numbers have unique properties that affect how we solve inequalities involving them. Here are a few important principles:

• When you multiply or divide by a negative number, the direction of the inequality sign must change to maintain a true statement.
• Adding or subtracting a negative number is straightforward; add or subtract just like positive numbers without altering the inequality sign.

These rules are crucial because they ensure that the inequality accurately reflects the order of values in relation to zero, which is the common axis for comparing positive and negative numbers. Always keep this in mind: the rules for negative numbers help maintain the balance and accuracy of inequalities.