Chapter 11.2 More Properties of Inequalities & Chapter 11.3 Simple Linear Inequalities

1. For any two numbers, and b, one and only one of the following relationships holds:

• a < b
• a = b
• a > b

2. If a < b and b < c, then a < c

3.  When a number is added to or subtracted from both sides of an inequality, the inequality holds.

• If a < b, then a + k < b + k

4. When both sides of an inequality are multiplied by a non-zero number k, the inequality holds for k > 0, but the inequality sign is reversed for k < 0.

• if a < b and k >0, then ka < kb
• if a < b and k < 0, then ka > kb

5. We can apply the properties of inequalities learned in the previous sections to solve simple linear inequalities in one variable, such as

• 3x + 5 < 17
• 4x -9 > 7x + 8
• represents the solution on a number line

6. HW assignment

• handout: 1 page
• workbook: page 18:#7, #8, #9; page 19: #14, #15, #16, #17, #18

1. For any two numbers, and b, one and only one of the following relationships holds:

• a < b
• a = b
• a > b

2. If a < b and b < c, then a < c

3.  When a number is added to or subtracted from both sides of an inequality, the inequality holds.

• If a < b, then a + k < b + k

4. When both sides of an inequality are multiplied by a non-zero number k, the inequality holds for k > 0, but the inequality sign is reversed for k < 0.

• if a < b and k >0, then ka < kb
• if a < b and k < 0, then ka > kb

5. We can apply the properties of inequalities learned in the previous sections to solve simple linear inequalities in one variable, such as

• 3x + 5 < 17
• 4x -9 > 7x + 8
• represents the solution on a number line

6. HW assignment

• handout: 1 page
• workbook: page 18:#7, #8, #9; page 19: #14, #15, #16, #17, #18

Chapter 11 Inequalities

Chapter 11.1 Solving Simple Inequalities
1. Idea of inequality

• An inequality is a statement that indicates that one quantity is less than or greater than another. You can write inequalities using symbols such as ” <“, “> “, “<=”, or “>=”.
• 3x < 10
• x < 10/3 All values of x that satisfy the inequality above are called the solutions of the inequality.

2. Solving an inequality

• In class, explore the properties of simple inequalities.
• If a > b and k > 0, then ka > kb
• if a < b, and k > 0, then ka < kb
• Lot’s of examples and exencise in class.

3. HW:

• Handout: 2 pages
• Workbook: page 17, #1, #2, #3, #4; page 18, #5, #6

Exam: coordinates and linear graphs.

Review materials and exam page are in the Google Classroom.

10.3 Slopes of Linear Graphs

1. Slope: the measure of the steepness of a straight line.

The slope of a line is the measure of the ratio of vertical rise and horizontal run.

The slope of a line = Rise/Run = Vertical change/Horizontal change

Positive slope, negative slope, special slopes for horizontal line and vertical line

2. The slope for any line with two points on the line A(x1, y1) and B(x2, y2)

The slope of a line = Rise/Run = (y2 – y1)/(x2 – x1)

Lots of examples in class.

HW:

• Handout
  • 2 pages
• Workbook:
  • page 11: 9, 10;
  • page 12: 11;
  • page 13: 17

10.2 Linear Graphs

1. General Form

The graph of an equation of the form y = mx + b is a straight line, where b and m are constant.

The graph of an equation that is a straight line is called a linear graph.

2. Special linear graph:

• y = c   — a horizontal straight line that is parallel to the x-axis
• x = a   — a vertical straight line which is parallel to the y-axis

 3. How to draw the graph of y = mx + b

• y = – (1/3)x + 2
• How to create a graph of the linear equation y = 2x + 7

• How to create a graph of the linear equation 5x + 2y = 20

4. Does a point A (x, y) lie on the graph?

5. Homework:

• Handout: 2 pages
• Workbook: page 10. #4, 5, 6, 7, 8

Chapter 10 Coordinates and Linear Graphs

10.1 Cartesian Coordinate System

1. Terminologies: x-axis, y-axis, coordinate axes, origin, first, second, third, and fourth quadrants. The coordinate plane, Cartesian plane, or x-y plane. 

The position of any point, say P, on the plane can be described by an ordered pair (a, b) of real numbers. We obtain the value of a when the vertical line through P intersects the x-axis and the value of b when the horizontal line through P intercepts the y-axis. We call a the x-coordinate and b the y-coordinate of P. We say that P has coordinates (a, b) and refer the point P as P(a, b). Hence, we can locate a point in a plane if we know its x- and y- coordinates.

2. Lots of exercise of stating the coordinates of points in a diagram, and plotting points with coordinates on a coordinate plane.

3. Home Work

• Handout: 2 pages
• Workbook: 

                     page 9: 1, 2

                     page 10: 3

The exam will cover all the materials we have learned in the Fall 2021-2022 term, namely chapters 3. 4. 5 & 8. See course info here.

The exam will be posted Saturday night in Google Classroom. Please print out all the pages before class.

1. Perpendicular Bisectors and Angle Bisectors

• Use of compass: center, radius, pin leg, and drawing leg
• To draw a circle
• To mark off or copy a line segment
• How to draw a perpendicular bisector of a line segment?
• Any point on the perpendicular bisector of a line segment is equidistant from the two  end points of the segment.

2. Angle Bisectors

• A ray AZ divides <BAC into two equal angles, <BAZ and <CAZ. The ray is called the angle bisector of <ABC
• How to draw an angle bisector?
• Any point on the angle bisector of an angle is equidistant from the two sides of the angle.

3. Class work

• Construct/draw circles, triangles, angles, equal line segments
• construct /draw perpendicular bisectors of line segments
• construct/draw angle bisectors of angles

4. Classification of Triangles

• The number of equal sides in the triangle: scalene triangle – no equal sides; isosceles triangle – two equal sides; equilateral triangles – three equal sides
• The type of angles of the triangle: acute-angled triangle – all angles are acute; right-angled triangle – one of the angles is a right angle; obtuse-angled triangle – one of the angles is an obtuse angle
• Is an equilateral triangle also an isosceles triangle?
• Is it Possible to draw a triangle with more than one obtuse angle?
• Can a scalene triangle be an acute-angled, right-angled or obtuse-angled triangle?
• All the three angles in a scalene triangle are different size
• The angles opposite the equal sides of an isosceles triangle are equal
• All the three angles in an equilateral triangle are equal in size

5. Quadrilaterals

• A closed plane figure with four straight sides joined by four vertices is called a quadrilateral
• Vertices, diagonals
• Properties of special quadrilaterals
• Parallelogram: 2 pairs of parallel and equal opposite sides
• Rectangle: all angles are right angles
• Rhombus: all sides are equal, diagonals are perpendicular to each other
• Square: all sides are equal, all angles are right angles
• Trapezoid: 1 pair of parallel sides

6. Home Work:

• Handout:
  • Two pages
• Workbook:
  • Page 48-50: 12, 13, 14,15,16

Chapter 8. Angles, triangles and quadrilaterials

1. Points, Line and Planes

• Point: has position; has no size
• Line: has an infinite number of points; has no width; can be determined by two points; can be straight or curved
• Ray: a part of a line with one endpoint
• Endpoint
• Line segment: a part of a line between two end points; has length
• Plane: a flat surface; has no thickness
• Parallel lines: two lines on the same plane do not intersect (meet or cut)
• Perpendicular (lines) to each other: two lines intersect at right angle
• Foot of the perpendicular

2. Types of angles

• Acute angle: angle < 90 degree
• Right angle: angle = 90 degree
• Obtuse angle: 90 degree < angle < 180 degree
• Reflex angle: 180 degree < angle < 360 degree

3. Complementary, supplementary, and adjacent angles

• Complementary angles: the sum of two angles is 90 degree
• Supplementary angles: the sum of two angels is 180 degree
• Adjacent angles: two angles share a common side and a common vertex but do not overlap

4. Properties of Angles

• The sum of adjacent angles on a straight line is 180 degree
• The sum of all angles at a point is 360 degree
• Vertically opposite angles: when two lines intersect, the vertically opposite angles are equal

5. Home Work:

• Handout:
  • Two pages
• Workbook:
  • Page 45: 1, 2, 3
  • Page 46: 4, 5

Exam: all about equations

1. Solve equations

2. Constructing/Forming Linear Equations to Solve Problems

The steps involved in problem-solving with linear equations are:

• Step 1. Read the question carefully and identify the unknown quantity
• Step 2. Use a letter to represent the unknown quantity (e.g. x)
• Step 3. Express other quantities in terms of x
• Step 4. Construct/Form an equation based on the given information
• Step 5. Solve the equation
• Step 6. Write down the answer statement