Download How to solve inequalities and apply the distance formula

How to solve inequalities and apply the distance formula
Real Number System and Interval Notation: All the real numbers can be represented by points
on a straight line (the real line). One distinguished point on the line represents the quantity 0, and a
unit length is given so that k units from the point 0 and on the right side of 0 represents the positive
integer k, while its mirror image on the left side of 0 represents the negative integer k, and so on. We
use R to denote the set of all real numbers.
Intervals are used to denote subsets of the real numbers. For real numbers a and b with a < b,
[a, b] = {x ∈ R : a ≤ x ≤ b} = {x : x is a real number with (a ≤ x ≤ b}.
(a, b) = {x ∈ R : a < x < b} = {x : x is a real number with (a < x < b}.
We call [a, b] a closed interval and (a, b) and open interval. Similarly, we can also define half open,
half closed intervals:
[a, b) = {x ∈ R : a ≤ x < b} and (a, b] = {x ∈ R : a < x ≤ b}.
The following notations are also used:
[a, ∞) = {x ∈ R : a ≤ x}
and
(a, ∞) = {x ∈ R : a < x}
(−∞, a] = {x ∈ R : x ≤ a}
and
(−∞, a)) = {x ∈ R : x < a}
Examples of solving linear inequalities
Example 1 Solve the linear inequality 4x + 1 < −5.
Example 2 Solve −7 < 4x + 1 < −5.
Example 3 Solve x2 + 3x − 4 > 0.
Absolute Values, and Distance Formula in The
of x is


 x
|x| =
0

 −x
Plane For a real number x, the absolute value
if x > 0
if x = 0
if x < 0
and,
|x| < a ⇐⇒ −a < x < a.
For two real numbers x and y, |x − y| represents the distance between x and y, and
|x − b| < a ⇐⇒ b − a < x < b + a.
Example 5 Solve |x − 3| < 4.
The Cartesian Plane and The Distance Formula Two perpendicular real lines can form a Cartesian
plane. Traditionally, the horizontal real line is the x-axis, and the vertical real line if the y-axis. Any
point on the plane uniquely corresponds to an ordered pair (x, y), refereed as the coordinates of the
point. The distance between the points (x1 , y1 ) and (x2 , y2 ) on the plane is
p
d = (x1 − x2 )2 + (y1 − y2 )2 .
Example 6 Find the distance between the points (−1, −2) and (3, 2).
1