1. COORDINATES, GRAPHS AND LINES
RAFIZAH KECHIL, UiTM PULAU PINANG
(Source: Cengage Learning,Hasfazilah Ahmat)

2. At the end of this lesson, students should be able to:
• Find the absolute values of real.
• Solve inequalities involving absolute values.

3. Absolute Value : Definition of Absolute Value
The absolute value represent magnitude, or the distance between the origin and the point on the real number line.

4. Example 7 – Finding Absolute Values
c. | – 4.3 | = 4.3
d. –| – 6 | = – (6)
= –6

5. Properties of Absolute Value

6. Inequalities Involving Absolute Values

7. Example 5 – Solving an Absolute Value Inequality
Solve each inequality.
a. | x – 5 | < 2
Solution:
a. | x – 5 | < 2
–2 < x – 5 < 2
–2 + 5 < x – < 2 + 5
3 < x < 7
The solution set is all real numbers that are greater than 3 and less than 7, which is denoted by (3, 7).
Write original inequality.
Write equivalent inequalities.
| x – 5 | < 2: Solutions lie inside (3, 7).
Add 5 to each part.
Simplify.

8. Example 5 – Solution b. | x + 3 |  7 x + 3  –7 or x + 3  7
cont’d
b. | x + 3 |  7
x + 3  – or x + 3  7
x + 3 – 3  –7 – x + 3 – 3  7 – 3
x  –10 x  4
The interval notation for this solution set is (-∞ ,–10][4,∞).
The symbol  is called a union symbol and is used to denote the combining of two sets.
Write original inequality.
Write equivalent inequalities.
Subtract 3 from each side.
Simplify.
| x + 3 |  7: Solutions lie outside (–10, 4).

9. Solving Linear

10. Solving Linear

11. Solving Rational

14. At the end of this lesson, students should be able to:
Use the imaginary unit i to write complex numbers.
Add, subtract, and multiply complex numbers.
Use complex conjugates to write the quotient of two complex numbers in standard form.
Find complex solutions of quadratic equations.

15. Complex Numbers The imaginary unit i, defined as i = where i 2 = –1.

16. Complex Numbers
The set of real numbers is a subset of the set of complex numbers, as shown in Figure 1.
Figure 1

17. Complex Numbers : Algebraic Operation

18. Example 1 – Adding and Subtracting Complex Numbers
a. (4 + 7i ) + (1 – 6i ) = 4 + 7i + 1 – 6i
= (4 + 1) + (7i – 6i )
= 5 + i
b. (1 + 2i ) – (4 + 2i ) = 1 + 2i – 4 – 2i
= (1 – 4) + (2i – 2i )
= –3 + 0
= –3
Remove parentheses.
Group like terms.
Write in standard form.
Remove parentheses.
Group like terms.
Simplify.
Write in standard form.

19. Example 1 – Adding and Subtracting Complex Numbers
cont’d
c. 3i – (–2 + 3i ) – (2 + 5i ) = 3i + 2 – 3i – 2 – 5i
= (2 – 2) + (3i – 3i – 5i )
= 0 – 5i
= – 5i
d. (3 + 2i ) + (4 – i ) – (7 + i ) = 3 + 2i + 4 – i – 7 – i
= (3 + 4 – 7) + (2i – i – i )
= 0 + 0i
= 0

20. Complex Numbers
Properties used when two complex numbers are multiplied.
(a + bi )(c + di ) = a(c + di ) + bi(c + di )
= ac + (ad )i + (bc)i + (bd )i 2
= ac + (ad )i + (bc)i + (bd )(–1)
= ac – bd + (ad )i + (bc)i
= (ac – bd ) + (ad + bc)i
Distributive Property
Distributive Property
i 2 = –1
Commutative Property
Associative Property

21. Example 2 – Multiplying Complex Numbers
a. 4(–2 + 3i ) = 4(–2) + 4(3i )
= – i
b. (2 – i )(4 + 3i ) = 2(4 + 3i ) – i(4 + 3i )
= 8 + 6i – 4i – 3i 2
= 8 + 6i – 4i – 3(–1)
= (8 + 3) +(6i – 4i )
= i
Distributive Property
Simplify.
Distributive Property
Distributive Property
i 2 = –1
Group like terms.
Write in standard form.

22. Example 2 – Multiplying Complex Numbers
cont’d
c. (3 + 2i )(3 – 2i ) = 3(3 – 2i ) + 2i(3 – 2i )
= 9 – 6i + 6i – 4i 2
= 9 – 6i + 6i – 4(–1)
= 9 + 4
= 13
The product of two complex numbers can be a real number.
Distributive Property
Distributive Property
i 2 = –1
Simplify.
Write in standard form.

23. Example 2 – Multiplying Complex Numbers
cont’d
d. (3 + 2i )2 = (3 + 2i )(3 + 2i )
= 3(3 + 2i ) + 2i(3 + 2i )
= 9 + 6i + 6i + 4i 2
= 9 + 6i + 6i + 4(–1)
= i – 4
= i
Square of a binomial
Distributive Property
Distributive Property
i 2 = –1
Simplify.
Write in standard form.

26. Complex Numbers: Complex Conjugates
This pairs of complex numbers of the form a + bi and a – bi, called complex conjugates.
(a + bi )(a – bi ) = a2 – abi + abi – b2i 2
= a2 – b2(–1)
= a2 + b 2

27. Complex Numbers: Complex Conjugates
To write the quotient of a + bi and c + di in standard form, multiply the numerator and denominator by the complex conjugate of the denominator to obtain
Standard form

28. Example 4 – Writing a Quotient of Complex Numbers in Standard Form
Multiply numerator and denominator by complex conjugate of denominator.
Expand.
i 2 = –1
Simplify.
Write in standard form.

29. Complex Solutions of Quadratic Equations
When solving a quadratic equation, you often obtain a result such as , which is not a real number. By factoring out i = , we can write this number in standard form.
The number i is called the principal square root of –3.

30. Example 6 – Complex Solutions of a Quadratic Equation
Solve (a) x2 + 4 = 0 and (b) 3x2 – 2x + 5 = 0.
Solution: a. x2 + 4 = 0
x2 = –4
x = 2i
b. 3x2 – 2x + 5 = 0
Write original equation.
Subtract 4 from each side.
Extract square roots.
Write original equation.
Quadratic Formula

31. Example 6 – Solution cont’d Simplify. Write in standard form.