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The addition principle is one of the foundational tools used in solving inequalities and equations. It tells us that we can add or subtract the same number from both sides of an equation or inequality without changing its solution. This principle maintains the balance of the inequality, ensuring that the relationship between the two expressions remains constant.For example, consider the inequality given: 7 - 8y > 5 - 7y In the first step of the solution, the principle is used to move all variable terms to one side. By adding 7y to both sides, the equation becomes:7 - 8y + 7y > 5 - 7y + 7yWhich simplifies to:7 - y > 5This step makes it easier to isolate the variable in the following steps. Remember, using the addition principle correctly can simplify complex inequalities and help you see the solution more clearly.

The multiplication principle is another critical concept for solving inequalities. It states that you can multiply or divide both sides of an inequality by the same non-zero number. However, with inequalities, there's an additional rule to remember: if you multiply or divide by a negative number, you must reverse the inequality sign.Consider our inequality at the final step:-y > -2Here, to solve for y, we multiply both sides by -1. Since we're multiplying by a negative number, we must reverse the inequality sign:-1 * (-y) < -1 * (-2)This simplifies to:y < 2Without reversing the inequality, the solution would be incorrect. Always keep this rule in mind when dealing with inequalities to ensure you find the correct solution.

One of the primary goals when solving inequalities is to isolate the variable on one side of the inequality. This means getting the variable by itself, usually on the left side, so you can easily determine the range of its possible values.In our example, after using the addition principle to combine variable terms, we had:7 - y > 5Next, we needed to isolate y. We did this by subtracting 7 from both sides:7 - 7 - y > 5 - 7Which simplifies to:-y > -2After isolating the term -y, we then used the multiplication principle to solve for y by multiplying both sides by -1 and reversing the inequality sign, resulting in:y < 2By following these steps, you can systematically isolate and solve for the variable in any inequality.