Class Notes (838,386)
Mathematics (1,919)
MATH 135 (334)
John Paul (2)
Lecture

# Optional_Equalities_and_Equations.pdf

2 Pages
48 Views

School
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
More Less

Related notes for MATH 135
Me

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.