SOLUTION: What role of operations that applies when you are solving an equation has a major exception when you are solving an inequality? Why?

Question 4286: What role of operations that applies when you are solving an equation has a major exception when you are solving an inequality? Why?
Answer by rapaljer(4671) (Show Source): You can put this solution on YOUR website!
I'm not sure if I understand your question, but let me try to answer it anyway.

When solving equations, I always teach the "Golden Rule of Mathematics," which is "Do unto one side of the equation as you do unto the other!" In other words when you add a number to one side of an equation, you must add that same number to the other side of the equation in order to keep the equation "in balance"--that is, keep it "equal." Likewise, when you subtract a number from one side of an equation, you subtract that same number from the other side as well. When you multiply or divide one side of an equation by a NON-ZERO number, then you must multiply or divide the other side by the same number.

When dealing with inequalities, the rule about adding and subtracting numbers from each side of an inequality is the same. However, What is different is that when you multiply or divide both sides of an inequality by a NEGATIVE number, then you must REVERSE the inequality sign.

Consider the following examples for operations with inequalities:

2 < 16 True statement. Now add 4 to each side.
2+4 < 16 + 4
6 < 20 Still true. Now subtract 8 from each side.
6-8 < 20-8
-2 < 12 Still true. Now multiply each side by 3.
3(-2) < 3(12)
-6 < 36 Still true. Now divide each side by 2.
-6/2 < 36/2
-3 < 18 Still true.

However, watch this!! Start with

-3 < 18 True statement. Now multiply (or divide!) each side by -1.
-1(-3)< -1(18)
3 < -18? FALSE!!

What happens, is when you multiply or divide by a negative number, it reverses the direction of the inequality. What WAS positive becomes negative, and what WAS negative becomes positive.

R^2 from SCC

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