91Ó°ÊÓ

The addition principle is a fundamental concept in solving inequalities and equations. It states that if you add (or subtract) the same number from both sides of an inequality or equation, the inequality or equation remains balanced.
For instance, consider an inequality like:
6 - 4y > 4 - 3y
To simplify this, we might want to get all the variable terms on one side.
In this case, we added 3y to both sides:
6 - 4y + 3y > 4 - 3y + 3y
This simplifies to:
6 - y > 4
Adding the same value (or subtracting) ensures the original relationship between the two sides is maintained.
This principle is crucial to simplifying inequalities step-by-step.

The multiplication principle is another key tool in solving inequalities. This principle asserts that if you multiply or divide both sides of an inequality by a positive number, the inequality direction remains the same.
However, if you multiply or divide by a negative number, you must reverse the direction of the inequality.
This is critical for maintaining the inequality's accuracy.
Let's consider the inequality example from earlier after we simplified it to:
-y > -2
To isolate the variable y, we multiply both sides by -1.
Remember, since we are multiplying by a negative number, we reverse the inequality:
-1(-y) < -1(-2)
Which simplifies to:
y < 2
The multiplication principle helps in effectively isolating and solving for the desired variable.

Isolating the variable is the process of rearranging an inequality or equation so that the variable is on one side by itself.
This often involves a combination of addition, subtraction, multiplication, and division.
In the given example, we started with:
6 - 4y > 4 - 3y
By using the addition principle, we moved all variable terms to one side:
6 - 4y + 3y > 4 - 3y + 3y
This simplifies to:
6 - y > 4
Then, using subtraction, we moved constants to the other side:
6 - y - 6 > 4 - 6
This simplifies to:
-y > -2
Finally, using the multiplication principle to handle the negative sign, we got:
y < 2
Isolating the variable step-by-step ensures accuracy and makes solving the inequality straightforward.