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Chapter 1. Inequalities. Section 1.1. Introduction. The Set of Real Numbers. The set of real numbers satisfies several important properties under addition and multiplication: e.g., closure, commutativity, associativity, distributive property.

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Inequalities

Section 1.1

Introduction

The Set of Real Numbers
• The set of real numbers satisfies several important properties under addition and multiplication: e.g., closure, commutativity, associativity, distributive property.
• The sum and product of two positive numbers is positive.
• The real number a is negative if and only if –a is positive.
• The set of real numbers is an ordered set.
Definition
• A statement that two numbers are not equal is called an inequality.
• For any two real numbers a,b, a > b if and only if a – b > 0. We say that b > a if and only if b – a > 0.
• Let a,b be real numbers. We write a  b if either a > b or a = b. We write a  b if either a < b or a = b.
Theorem (Law of Trichotomy)
• For any real numbers a and b, exactly one of the following is true:

a = b, a > b, a < b.

Theorem*
• Let a and b be real numbers such that a
• If c is any real number, then a+c < b+c.
• If c is a positive number, then ac bc.
Theorem* (Transitive Property)
• Let a, b, c be real numbers such that a < b and b < c. Then a < c.
Example 1
• Let a and b be real numbers such that a < b. Show that
Example 2
• Let a and b be real numbers with a > b > 0. Prove that a(2a + b) > b2.
Example 3
• Let a and b be real numbers such that a < b < 1. Prove that a + ab < a2 + b.
Example 4
• Let a and b be positive numbers. Prove that a > b if and only if a2 > b2.
Theorem*
• If a is any real number, a2 0.
Example 1
• Let a and b be real numbers. Show that
Example 2 (AM-GM Inequality)
• Let a and b be nonnegative real numbers. Prove that
Example 3
• Let a, b, x, y be positive real numbers such that x2 + y2 = 1 and a2 + b2 = 1. Prove that ax + by  1.
Exercises / Assignment Items
• Prove the following statements:
• If a > b and c > d, then a + c > b + d.
• If 0 < a < 1, then a2 < a.
• If a < b < c, then .
• For any real numbers a, b, c, and d, (ac + bd)2 (a2 + b2)(c2 + d2).
• For 0 < a < b, let h be defined by . Then a < h < b.

Section 1.2

Polynomial and Rational Inequalities

Definition
• The domain of a variable in an inequality is the set of real numbers for which both sides of the inequality is defined.
Examples
• The inequality 3x3 4x2 + 7x has the set of all real numbers as its domain.
• The inequality

has the set R \ {2008} as domain.

Remark
• There are cases when all elements in the domain of a variable satisfy the inequality.
• x + 2 < x + 5
• x2 + 5  0
• However, in general, not all members in the domain of a variable in an inequality yields a true inequality when substituted to the variable.
Definition
• Any member in the domain of a variable for which the inequality is true after substitution into the variable is a solution of the inequality.
• The set of all solutions is called the solution set of the inequality.
Types of Inequalities
• If the solution set of an inequality is exactly the same as the domain of the variable, then we say the inequality is absolute.
• A conditional inequality is one for which there is at least one member in the domain of the variable that is not in the solution set of the inequality.
Writing Sets of Real Numbers
• Set builder notation

Example: {x|x1}

• Interval notation

Example: [-1,+)

Examples
• Solve the following inequalities, and express the final answer in interval notation:

(1) 5x + 6  x + 2

(2)

(3) x2 (x + 2) > (4x + 5)(x + 2)

(4) 2x2 + 3x – 1 < 6 – 2x  3x + 2

(5)

(6)

(7)

Assignment Items
• Solve the ff. inequalities, and express the final answer in interval notation:

(1)

(2)

(3) (3 – 4x)2006(x + 1)2007(7 – 2x)2008 0

(4)

(5)

Section 1.3

Equations and Inequalities Involving the Absolute Value

Definition
• The absolute value of a real number is the distance between 0 and the number on the real line.
• If x is a real number, then

if x  0

if x < 0

• Note that this is the same as the definition of the principal square root of x2. That is, .
Theorem
• Let a and b be real numbers. Then
• |ab| = |a||b|
• if b  0
Remark
• Although we can “split” the absolute value of a product or quotient, the same cannot be said for the sum or difference of real numbers. That is, |a  b|  |a|  |b|.
Examples
• Find the solution set:

(1) |3 – 8x| = 13

(2) 2|3 – 2x| = 5|x + 1|

(3) |4 – |6 – 7x|| = 9

(4) |x – 3| + |x – 2| + |1 – x| = 3

(5) |5 – 3x – |3x + 1|| – 4 = –2x

Theorem

Let a > 0. Then

• |x| < a if and only if –a
• |x| > a if and only if x<–a or x>a.
• |x|  a if and only if –axa.
• |x|  a if and only if x–a or xa.
Examples
• Solve the following inequalities:

(1)

(2)

(3)

(4)

Lemma*
• For any real number x, the following inequality is true:
Theorem* (Triangle Inequality)
• If a and b are real numbers, then

|a + b|  |a| + |b|

Example
• Suppose that x and y are real numbers such that |x – 1|  3 and |y + 2|  1. Prove that |3y – 2x|  17.
Exercises / Assignment Items
• Find the solution set. For inequalities, express your final answer in interval notation.

(1) |4 – 11x| = |5x – 28|

(2) |3 – 2x| – |x + 5| – |4 – x| = -8

(3) (5)

(4) (6)

• Suppose that |x + 5|  4 and |y – 2|  7. Show that |x + 2y|  19.

Chapter 2

Circles and Lines

Section 2.1

The Rectangular Coordinate System

Definition
• An ordered pair (x,y) of real numbers has x as its first member and y as its second member.
• The model of representing ordered pairs is called the rectangular coordinate system or the cartesian plane. It is developed by considering two real lines intersecting at right angles.
Definition
• Each point in the plane is identified by an ordered pair (x,y) of real number x and y, called the coordinates of the point.
• The first coordinate is the x-coordinate or abscissa and the second coordinate is the y-coordinate or ordinate.
Problem
• What is the distance between two points (x1, y1) and (x2, y2) in the plane?
Distance Between Two Points
• If the points lie on a horizontal line, y1 = y2, and the distance between the points is |x2 – x1|.
• If the points lie on a vertical line, x1 = x2, and the distance between the points is |y2 – y1|.
• If the two points do not lie on a horizontal or vertical line, they can be used to form a right triangle.
Theorem (Distance Formula)
• The distance d between the points A(x1,y1) and B(x2,y2) in the plane is given by
Example 1
• Show that the points A(2,1), B(4,0), and C(5,7) form the vertices of a right triangle.
Example 2
• Find the point on the y-axis that is equidistant from (-5,-2) and (3,2).
Theorem (Triangle Inequality)
• If P1, P2, P3 are any three points on the plane, then

Moreover, equality is satisfied if and only if P2 is a point on the line segment .

Example
• Determine whether the point (-1,0) is on the line segment joining (-9,-2) and (11,3).
Theorem (The Midpoint Formula)
• If M(x,y) is the midpoint of the line segment from A(x1,y1) to B(x2,y2), then

and .

Example 1
• One endpoint of a line segment is (8,1) and its midpoint is (3,7). Find the other endpoint.
Example 2
• Show that the triangle with vertices A(-3,-6), B(3,2), and C(5,0) is isosceles. Find its area.
Example 3
• Given A(-1,1) and B(3,-2), find the point 3/7 of the way from A to B.
Example 4
• Prove analytically that the diagonals of a parallelogram bisect each other.
Exercises / Assignment Items
• Ex. 2.1, #s 5, 9, 11, 14, 21, 23
• Let P(-1/2,0) and Q(1/2,0) be adjacent vertices of a regular hexagon. If the hexagon is below the segment PQ, find the coordinates of the four other vertices.

Section 2.2

Circles

Definition
• A circle is the set of all points in a plane a fixed distance from a fixed point. The fixed point is called the center and the fixed distance is called the radius of the circle.
Theorem (Standard equation of a circle)
• An equation of the circle with radius r and center at (h,k) is given by

(x-h)2 + (y-k)2 = r2

• This is often called the center-radius form or the standard form of the equation of the circle.
Definition
• A diameter of a circle is a line segment passing through the center which connects two points of the circle.
Example
• Find an equation of a circle whose diameter has endpoints (7,-3) and (1,7).
Remark
• If we expand the equation of a circle in standard form, and then group the terms, we can write the equation in its general form:

x2+y2+Dx+Ey+F=0

where D, E, and F are constants.

• Given the equation of a circle in general form, we can find its center and radius by completing the squares.
Example 1
• Determine the center and radius of the circle 3x2+3y2+12x+30y+45=0. Then sketch a graph of the circle.
Example 2
• For what values of r and s will x2 + y2 + rx + sy = 25 be the equation of a circle having center at (3,4)? Find the radius of this circle.
Example 3
• Find the equation of the circle passing through the points (2,8), (6,4), and (2,0).
Circle, point, or null set?
• Not all equations of the form x2+y2+Dx+Ey+F=0 are those of circles. To determine whether the equation is that of a circle, we write the equation in its center-radius form (x-h)2+(y-k)2 = a.
• If a>0, then the graph is a circle.
• If a=0, then the graph is a single point, namely, (-1/2 D, -1/2 E).
• If a<0, then the graph is a null set.
Example
• For what values of k is the equation

x2+y2+2x-4y+26=k2+4k

that of

(a) a circle?

(b) a single point?

(c) an empty set?

Exercises / Assignment Items
• Ex. 2.2, #9, 11, 14, 17, 23, 27, 28, 33, 35

Section 2.3

Lines

Definition
• If two points (x1,y1) and (x2,y2) are on a line L, then the slope m of the line L is defined by

, x2 x1.

The slope of a vertical line is undefined.

The Slope and the Direction of a Line
• If m>0, then as the value of x increases, the value of y also increases.
• If m<0, then as the value of x increases, the value of y decreases.
• If m=0, then the line is horizontal.
• The slope of a vertical line is undefined.
Example 1
• The slope of a line segment is 2/3 and one endpoint is (-1,4). If the other endpoint is on the x-axis, what are its coordinates?
Example 2
• Find the value(s) of t so that the points A(t-1, 2t-1), B(4,1-2t), and C(-3,3t+5) are collinear.
Point-Slope Form
• An equation of a line with slope m and passing through the point (x0,y0) is
Slope-Intercept Form
• An equation of a line with slope m and with y-intercept b is

y = mx + b.

Example 1
• Find an equation of the line passing through the points (-3,7) and (8,-9).
Example 2
• Find an equation of the line having the same y-intercept as 2x + 5y = -25 and twice the slope of 9x – 3y = 4.
Example 3
• Determine the value(s) of k in the equation 2x+3y+k = 0 so that this line will form a right triangle with the coordinate axes whose area is 27 square units.
Example 4
• The product of the x and y-intercepts of a line is -1 and the line passes through (-2,-6). Find an equation of this line, and the area of the triangle formed by the line with the coordinate axes.
Definition
• Two lines are parallel if they do not intersect, or equivalently, the distance between the two lines is a positive constant.
• Two lines are perpendicular if they intersect at right angles.
Theorem

Let l1 and l2 be two nonvertical lines with

slopes m1 and m2 respectively.

• The lines l1 and l2 are parallel if and only if m1=m2.
• The lines l1 and l2 are perpendicular if and only if m1m2=-1 or m2=-1/m1.
Example 1
• For what value(s) of k are the lines (k-1)x + y + 3 = 0 and (k+1)x – 3y + 5 = 0

(a) parallel?

(b) perpendicular?

Example 2
• The points A(-8,-16), B(0,10), and C(12,14) are three vertices of a parallelogram. Find the coordinates of the fourth vertex if it is located in the third quadrant.
Tangent Lines to a Circle
• A tangent line to a circle is a line that intersects the circle at exactly one point, called the point of tangency.
• The radius drawn to the point of tangency is perpendicular to the tangent line.
Example 1
• Find the equations (in slope-intercept form) of the lines tangent to the circle x2 + y2 – 8x + 10y – 128 = 0 at the points of the circle on the x-axis.
Example 2
• Find the equation of the circle that is tangent to the line 4x – 3y + 12 = 0 at the point (-3,0) and also tangent to the line 3x + 4y – 16 = 0 at the point (4,1).
Example 3
• Find the equation of the circle the passes through the points (2,1) and (3,5), and whose center is on the line 8x + 5y = 8.
Exercises / Assignment Items
• Ex. 2.3, #s 15, 25, 26, 27, 31, 35, 37, 43, 45