Connect and share knowledge within a single location that is structured and easy to search. We all learned in our early years that when dividing both sides by a negative number, we reverse the inequality sign. Why is the reversal of inequality?
What is going in terms of number line that will help me understand the concept better? Dividing an inequality by a positive number retains the same inequality.
But then if you switch side for all terms, each term faces the opposite "side" of inequality sign Moreover, it is not difficult to see that a strictly decreasing function reverses both types of non-strict inequalities. The same explanation can be used for taking the reciprocal of both sides of an inequality, when both sides are positive or when both sides are negative.
Similarly, strictly increasing functions preserve inequalities. This gives a sometimes useful application of the calculus task of determining on what interval s a function might be increasing or decreasing, by the way.
It looks like we're multiplying both sides by -1 and reversing the direction of the inequality. There's another way:. It's because negative numbers appear bigger but are actually smaller. We need to flip the sign to make it true. For negative numbers, things are flip-flopped.
NOTE : Multiplying by a negative is multiplying by a positive signs still same then multiplying by -1 Signs now flipped. Dividing is multiplying by a fraction, and dividing by a negative number is multiplying by a negative fraction. Also, if you think about it, this also holds when you reciprocate both sides. Sign up to join this community. The best answers are voted up and rise to the top. The output of an absolute value expression is always positive, but the " x " inside the absolute value signs might be negative, so we need to consider the case when x is negative.
That gives us our two inequalities or our "compound inequality". We can easily solve both of them. These kinds of problems take some practice, so don't worry if you aren't getting it at first!
Keep at it and it will eventually become second nature. You also often need to flip the inequality sign when solving inequalities with absolute values. How to Divide Negative Fractions. How to Solve Double Inequalities.
Standard Form of a Linear Equation. How to Solve Inequalities With Fractions. How to Divide Negative Numbers. How to Determine Linear Equations. The one on the left is less than the one on the right. The one on the right is greater than the one on the left. This is true whether the two numbers are both positive, both negative, a negative and a positive or zero and any number. When you multiply or divide any non-zero number by a negative, its sign changes. If it was positive it becomes negative and if it was negative it becomes positive.
On a number line, the number "flips" over to the other side of zero. When you multiply or divide both sides of an inequality this happens to both numbers, the one on the "less than" side and the one on the "greater than" side.
Both sides "flip". And when this happens the number that was on the left of the other before is now to the right of the other. In other words, what was less than before is now greater than. And the number that was to the right i. This is why we have to reverse "flip" the inequality whenever we multiply or divide it by a negative number.
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