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Optional_Equalities_and_Equations.pdf

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Department
Mathematics
Course
MATH 135
Professor
John Paul
Semester
Fall

Description
MATH 135 Fall 2013 Optional Reading I: Equalities and Inequalities of Real Numbers In mathematics, the same object may often be expressed in di▯erent forms. For example, the number 1 may be 2 p P 1 ▯1▯ j written as 2 or 4 ▯ 3 or as j=12 (Wow! where did that last one come from?), etc. However, no matter how we write it, the number 1 is just what it is. That is why, we use a relation known as equality (symbol: “=”) to convey the idea that regardless of the expressions, we may be talking about the same object. Definition 1. Given two expressions L and R, we write L = R and say L is equal to R (or that L and R are equal) to mean that both L and R represent the same number . When we write L = R, we say that L is the left hand side (L.H.S) and R is the right hand side (R.H.S) of the equation. Therefore, from the above discussion, 2 p X ▯1▯ j 1 = = 4 ▯ 3 = : 2 j=1 2 1 Manipulating Equations If you are given an equation L = R, then for any real number x, we may say: 1. L + x = R + x and L ▯ x = R ▯ x. 2. L = R + (x ▯ x) and L + (x ▯ x) = R. 3. L ▯ x = R ▯ x. 4. If x is not zero, thx= x. 5. If x is not zero, then L= R and L = R ▯ . x x 6. L ▯ R = 0 and 0 = R ▯ L. 7. If f is a well-defined function, then (as long as L and R come from the domain of f): f(L) = f(R): For example, if we have an equation ax + bx + c = 0; where a , 0, and b > 4ac then we may manipulate it to solve for x through the following steps: ax + bx + c = 0 =) ax + bx = ▯c subtracting c from both sides 2 b c =) x + a x = ▯a dividing both sides by a 2 b b2 b2 b2 =) x + x + = ▯c + adding to both sides a !a 4a 4a b 2 b2 =) x + = ▯c + completing the square 2a 4a r b b2 =) x + = ▯ ▯c + square root both sides 2a 4a r 2 b b b =) x = ▯ 2a ▯ ▯c + 4a subtracting 2a from both sides p ▯b ▯ b ▯ 4ac =) x = algebraic rearrangement. 2a 1 2 Proving Equalities Often we suspect two expressions to be the same, and are
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