Unizor - Algebra - Inequalities - Transformations

• Report rights infringement
• published: 20 May 2013
• views: 341

1. Addition and subtraction of the same number (positive, negative or zero) to or from both sides of inequality does not change the character of that inequality. It can be illustrated by using the following logic. If a is less than b then there exists a positive number D such that a + D = b. This is an equality. We can add any number X to both parts of it and the equality would hold: a + D + X = b + X. Since D is still the same positive number as above and addition is a commutative operation, we can write it as a + X + D = b + X. The above equality is a definition of a "less than" relationship a + X is less than b + X as we wanted to prove. 2. Multiplication and division represent a different case. Again, assume that a is less than b and there exists a positive number D such that a + D = b. Let's multiply this equality by the same number X. The result is (a + D)·X = b·X. We are interested in the relationship between a·X and b·X. Using the distributive law of multiplication and the equality above, we can state the following: a·X + D·X = b·X. The difference between a·X and b·X is D·X. If it's positive than indeed a·X is less than b·X. If it is negative, the character of inequality is reversed a·X is greater than b·X. Since we started with positive difference D between a and b, the sign of a difference D·X depends exclusively on the sign of a multiplier X. It's positive if X is greater than 0 and negative if X is less than 0. Summarizing the above, we derive with the following rule. Multiplication of both sides of an inequality by the same positive number does not change the character of the inequality; multiplication of both sides of an inequality by a negative number changes the sign of inequality to an opposite; obviously, multiplication by 0 transforms an inequality into a trivial identity 0 = 0. Absolutely the same logic is applicable to a division. Division by a positive number does not change the inequality; division by a negative number changes the inequality to an opposite. Zero is not a candidate for division, of course. 3. Absolute value taken from both sides of an inequality might change it in different ways depending on the numbers involved and there is no rule about it. -2 is less than 2 and |-2| = |2| 4. Application of monotonically increasing function to both sides of an inequality used as its arguments retains the relationship between the obtained values. The reason is that this is the definition of the monotonically increasing functions, that is if one argument is less than another then the value of a function applied to the first argument is less than the value of this function applied to the second argument. 5. Application of monotonically decreasing function to both sides of an inequality used as its arguments reverses the relationship between the obtained values. The reason is that this is the definition of the monotonically decreasing functions, that is if one argument is less than another then the value of a function applied to the first argument is greater than the value of this function applied to the second argument. 6. Now we can re-examine the results of the application of the absolute value to both sides of an inequality. If we know beforehand that both sides of an inequality are positive then the application of the absolute value to them does not change the inequality because the absolute value as a function is monotonically increasing among positive arguments. If we know beforehand that both sides of an inequality are negative then the application of the absolute value to them reverses the inequality because the absolute value as a function is monotonically decreasing among negative arguments. 7. Analogously to the absolute value, the application of a power function x^2 or any other power function with an even exponent (x^4, x^24 etc.) retains or reverses the inequality depending whether both sides of an inequality are positive (retains) or both are negative (reverses). 8. Inverse function 1/x has its own specifics. We know that it is monotonically decreasing on two separate intervals, for argument changing from minus infinity to 0 and from 0 to plus infinity. Therefore, if we know beforehand that both sides of an inequality are of the same sign (both positive or both negative) then inversing the inequality reverses the relationship, that is if p is less than q then 1/p is greater than 1/q. If we know beforehand that the sides of an inequality are of opposite signs (one is negative and, therefore, less than another, positive) then the relationship is retained because inverting the number does not change its sign, that is if p is less than 0, q is greater than 0 and, therefore, p is less than q then 1/p is less than 1/q because 1/p is less than 0 and 1/q is greater than 0.