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 ▯ .
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
=) x + = ▯ ▯c + square root both sides
b b b
=) x = ▯ 2a ▯ ▯c + 4a subtracting 2a from both sides
▯b ▯ b ▯ 4ac
=) x = algebraic rearrangement.
1 2 Proving Equalities
Often we suspect two expressions to be the same, and are