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### Chapter 1

### Section 1.1

### Section 1.2

### Section 1.3

### Chapter 2

### Section 2.1

### Section 2.2

### Section 2.3

Inequalities

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.

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.

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

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)

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 –axa.
- |x| a if and only if x–a or xa.

Lemma*

- For any real number x, the following inequality is true:

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.

Circles and Lines

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.

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

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

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

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