Chapter 2: Equations and Inequalities

MAT 1103: Fundamentals of Mathematics

2.1 Equations

1) Equation Statement indicating that 2 quantities are true. Example: Solution set:

3x − 2 = 10 values of variable that satisfy equations.

2) Restricted values a. Fraction:

1 , x≠a x−a

b. Radical: c. Logarithmic:

x−a, x≥a log( x − a) , x > a

3) Solve Linear equation in 1 variable

ax + b = 0
Example 1: Solve a.

(a and b are real numbers and a ≠ 0)

2x + 3 = 0

b. 3( x + 2) = 5 x + 2

c.

3y 2 1 − = y 2 3 5

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Chapter 2: Equations and Inequalities

MAT 1103: Fundamentals of Mathematics

4) Solve Rational Equations Example 2: Solve a.

3 7 + =2 5 x+2

b.

3x 5 − =3 x −1 x + 3

2.2 Applications of Linear Equations 1) English-mathematics vocabulary Mathematical operator + x English words More, greater, add, sum, exceeds, increase, higher, total, extra Less, difference, lower, minus, decrease, fewer Times, multiple

Mathematical ratio 2x 3x 1/3 x ¼x

English words Double Triple One third One quarter

2) General guideline for solving word problem a. Read the problem. b. Read the problem again. c. Draw a picture / table / flow chart.

d. Find and label the unknowns that you are looking for. e. Find and label the known quantities. f. Write down all the formulas and relations between the known and unknown.

g. Solve the problem. h. Check the answer & reply in words.

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Chapter 2: Equations and Inequalities

MAT 1103: Fundamentals of Mathematics

Number problems Example 1: The sum of three consecutive numbers is 78. Find the numbers.

Example 2: The sum of three consecutive odd numbers is 171. Find the numbers.

3) Geometric problems Example 3: The length of a rectangle is 4 cm longer than the width. If the perimeter is 96 cm, find the width.

2.3 Quadratic Equations

1) Quadratic equations •

ax 2 + bx + c = 0

a, b and c are real numbers and a ≠ 0

2) Zero Factor Theorem • If a and b are real numbers, and if ab = 0, then a = 0 or b = 0.

3) Factorization Example 1: Solve a. 2 x 2 − 9 x − 35 = 0 b. x 2 + 4 x = 0

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Chapter 2: Equations and Inequalities

MAT 1103: Fundamentals of Mathematics

4) Quadratic formula Given ax 2 + bx + c = 0 , Example 3: Solve a. x 2 − 5 x + 3 = 0 b. 2 x 2 + 8 x − 7 = 0

x=

− b ± b 2 − 4ac 2a

, a≠0

2.4 Inequalities 1) Symbols Meaning a is less than b a is greater than b a is less than or equal to b a is greater than or equal to b

ab a≤b a≥b

2) Number line Example 1: Draw the number line for each inequality a. Black dot for including. i.

x ≥ 2.5

ii. x ≤ 2.5

b. White dot for excluding. i.

x > 2.5

ii. x < 2.5

3) Properties • Let a, b and c represent real numbers

a. Addition and Subtraction ii. iii. If a < b , then a + c < b + c If a < b , then a − c < b − c

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Chapter 2: Equations and Inequalities

MAT 1103: Fundamentals of Mathematics

b. Multiplication and Division i. Keep the direction if multiply or divide with positive numbers.

a b - If a < b and c > 0 , then ca < cb and < c c ii. Reverse the direction if multiply or divide with negative numbers.

a b - If a < b and c < 0 , then ca > cb and > c c
4) Linear Inequalities In One Variable • Standard form

ax + c < 0 ax + c ≤ 0
Example 2: Solve a. 3( x + 2) < 8

ax + c > 0 ax + c ≥ 0

b. − 5( x − 2) ≤ 20 + x

c.

2 (x + 2) > 4 (x − 3) 3 5

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Chapter 2: Equations and Inequalities

MAT 1103: Fundamentals of Mathematics

5) Compound Inequalities in One Variable

Example 3: Solve a. 5 < 3 x − 7 ≤ 8 b. x + 3 ≤ 2 x − 1 < 4 x − 3

c. 3 + x ≤ 3 x + 1 < 7 x − 2

2.5 Absolute Values 1) Definition

x =

x , x≥0 −x , x 0, then If k ≥ 0, then |x| < k is equivalent to − k < x < k |x| < k is equivalent to − k ≤ x ≤ k

Example 3: Solve a. |x - 2| < 7 b. 2

8x + 2 ≤1 5

4) Inequalities of the Form |x| > k If k ≥ 0, then

x > k is equivalent to x < −k or x > k x ≥ k is equivalent to x ≤ −k or x ≥ k

Example 4: Solve a. 2 x − 7 − 3 > 2 b. 4 2 x + 3 + 1 ≥ 5

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