Real Numbers: are numbers that can be expressed as decimals, such
as 50, 30.346,-0.3456……
The real numbers can be represented geometrically as points on a
number line called the real line.
We distinguish three special subsets of real numbers.
1. The natural numbers, namely1, 2, 3, 4…….
2. The integers, namely±1,±2,±3,±4……..
3. The rational numbers, namely the numbers that can be expressed in
the form of a fraction , where m/n and n≠0are integers and Examples
are1/3,4/6,-9/20,3/1 ….....
finite Intervals: it is a subset of the real line that contains at least two
numbers and contains all the real numbers lying between any two of its
elements.
finite Closed Intervals: it is a finite interval that contains both of its
endpoints.
finite half-open Intervals: it is a finite interval that contains one
endpoint but not the other
finite open Intervals: it is a finite interval that contains neither
endpoint.
EXAMPLE 1 Solve the following inequalities and show their solution
sets on the real line.
Solution
The solution set is the open interval
The solution set is the open interval
.
.
(c)
The solution set is the half-open interval (1,11/5 ].
Absolute Value
The absolute value of a number x, denoted by is defined by
the formula
EXAMPLE 2Solving an Equation with Absolute Values
Solve the equation
EXAMPLE 3 Solving an Inequality Involving Absolute Values Solve the
inequality
The solution set is the open interval (1/3,1/2)
EXAMPLE 6 Solve the inequality and show the solution set on the real
line:
The solution set is the closed interval [1, 2].
Cartesian Coordinates in the Plane
If P is any point in the plane, it can be located by exactly one ordered
pair of real numbers in the
following way. Draw lines
through P perpendicular to the
two coordinate axes. These lines
intersect the axes at points with
coordinates a and b. The
ordered pair (a, b) is assigned to
the point P and is called its
coordinate pair. The first number
a is the x-coordinate of P; the
second number b is the ycoordinate of P. The xcoordinate of every point on the y-axis is 0. The y-coordinate of every
point on the x-axis is 0. The origin is the point (0, 0).
Pythagorean Theorem
For a right triangle with hypotenuse of length
and sides of lengths a and b , you have a2+b2=c2.
(The converse is also true. That is, ifa2+b2=c2,
then the triangle is a right triangle.)
Suppose you want to determine the distanced
between two points (x1,y1) and(x2,y2)in the plane.
With these two points, a right triangle can be
formed. The length of the vertical side of the triangle is
𝑦2 − 𝑦1 and the length ofthe horizontal side is 𝑥2 − 𝑥1 By the
Pythagorean Theorem, you can write
d2 = 𝑥2 − 𝑥1
2
+ 𝑦2 − 𝑦1
2
𝑥2 − 𝑥1
2
+ 𝑦2 − 𝑦1
2
d=
=
𝑥2 − 𝑥1
2
+ 𝑦2 − 𝑦1
2
The Distance Formula
The distance between the points (x1,y1) and (x2,y2)in the plane is
d=
𝒙𝟐 − 𝒙𝟏
𝟐
+ 𝒚𝟐 − 𝒚𝟏
𝟐
Example : Find the distance between the points (-2, 1) and (3, 4).
Solution Let (x1,y1) =(-2, 1) and(x2,y2 )= (3, 4)Then apply the
Distance Formula.
d=
𝒙𝟐 − 𝒙𝟏
=d=
𝟓
𝟐
𝟐
+ 𝒚𝟐 − 𝒚𝟏 𝟐 = d =
+ 𝟑
𝟐
𝟑 − (−𝟐)
𝟐
+ 𝟒−𝟏
𝟐
= d = 𝟑𝟒 ≈ 𝟓. 𝟖𝟑
The Midpoint Formula
The midpoint of the line segment joining the points(x 1,y1) and(x2,y2 ) is
given by the Midpoint Formula
Midpoint =
𝑥 1 +𝑥 2
2
,
𝑦1 +𝑦2
2
.
Example: Find the midpoint of the line segment joining the points
(-5,-3) and (9,3).
Solution Let (x1,y1) =(-5, -3) and(x2,y2 )= (9, 3).
Midpoint =
𝑥 1 +𝑥 2
2
=(2,0)
,
𝑦1 +𝑦2
2
= Midpoint =
−5+9
2
,
−3+3
2
DEFINITION Slope
The constant
is the slope of the non vertical line P1P2 .
We can write an equation for a non vertical straight line L if we know its
slope m and the coordinates of one pointP1(x1,y1) on it. If P(x, y) is any
other point on L, then we can use the two points P1 and P to compute
the slope,
The equation
is the point-slope equation of the line that passes through the point
(x1,y1) and has slope m.
EXAMPLE :Write an equation for the line through the point (2, 3) with
slope -3/2
Solution We substitute x1=2, y1=3 and m=-3/2 into the point-slope
equation and obtain
When x=0,y=6 so the line intersects the y-axis at y=6
EXAMPLE :A Line Through Two Points Write an equation for the line
through (-2,-1)and (3, 4).
Solution The line’s slope is
We can use this slope with either of the two given points in the pointslope equation:
The y-coordinate of the point where a non vertical line intersects the
y-axis is called the y-intercept of the line.
Similarly, the x-intercept of a non
horizontal line is the x-coordinate of the
point where it crosses the x-axis. A line
with slope m and y-intercept b passes
through the point (0, b), so it has equation
The equation is called the slope-intercept
equation of the line with slope m and yintercept b.
y = b + m(x – 0), or, more simply, y = m x + b.
The equation
y=mx+b
is called the slope-intercept equation of the line with slope m and yintercept b.
EXAMPLE :Finding the Slope and y-Intercept
Find the slope and y-intercept of the line 8x+5y=20
Solution Solve the equation for y to put it in slope-intercept form:
8x + 5y = 20
5y = 20 - 8x
y =4 – 8/5 x
The slope is m=-8/5 . The y-intercept is b=4
Parallel and Perpendicular Lines
Lines that are parallel have equal angles of inclination, so they have the
same slope (if they are not vertical). Conversely, lines with equal slopes
have equal angles of inclination and so are parallel.
If two non vertical L1lines L2 and are perpendicular, their slopes m1 and
m2 satisfy 𝑚1 𝑚2 = −1 so each slope is the negative reciprocal of the
other:
𝑚1 =
−1
−1
, 𝑚2 =
𝑚2
𝑚1
EXAMPLE : write an equation for each line described.
a. Passes through −2,1 and is parallel to the line 𝑥 + 2𝑦 = 1
b. Passes through 2,1 and is perpendicular to the line 𝑥 + 2𝑦 = 1
Solution
a. we find the slope of the line
𝑥 + 2𝑦 = 1 → 𝑦 = 1 2 − 1 2 𝑥 → 𝑚 = −1 2
Lines that are parallel have the same slope
equation of the line Passes through −2,1 and is parallel to the line
𝑥 + 2𝑦 = 1 is
y = 1 − 1 2x − 1 → y = −1 2x
b. we find the slope of the line
𝑥 + 2𝑦 = 1 → 𝑦 = 1 2 − 1 2 𝑥 → 𝑚 = −1 2
The slope of line perpendicular to the line 𝑥 + 2𝑦 = 1 𝑖𝑠 𝑚┴ = 2
perpendicular line is 𝑦 = 1 + 2𝑥 − 4 → 𝑦 = 2𝑥 − 3
H.W
Inequalities
solve the inequalities and show the solution sets on the real line.
Absolute Value
Solve the inequalities and expressing the solution sets as intervals or
unions of intervals. Also, show each solution set on the real line.
Distance and Midpoint
Find the distance and midpoint from A to B.
Slopes and Lines
find the slope of the line from A to B.
write an equation for each line described.
1. Passes through −1,1 with slope −1.
2. Passes through −2,3 with slope 1 2
3. Passes through (3, 4) and −2,5
4. Passes through −8,0 and −1,3
5. Has slope −5 4 and y-intercept 6
6. Has slope1 2 and y-intercept −1
7. Passes through −12,9 and has slope 0
8. Passes through (1 3 , 4), and has no slope
9. Has y-intercept 4 and x-intercept −1
10. Has y-intercept −6 and x-intercept 2
11. Passes through 5, −1 and is parallel to the line 2𝑥 + 5𝑦 = 15
12. Passes through −1,3 parallel to the line 2 𝑥 + 5𝑦 = 3
13. Passes through 4,10 and is perpendicular to the line 6𝑥 − 3𝑦 = 5
14. Passes through 0,1 and is perpendicular to the line 8𝑥 − 13𝑦 = 13