EQUATIONS, INEQUALITIES & ABSOLUTE VALUE. 2 CONTENT 2.1 Linear Equation 2.2 Quadratic Expression and Equations 2.3 Inequalities 2.4 Absolute value.

EQUATIONS, EQUATIONS, INEQUALITIES & INEQUALITIES &

ABSOLUTE VALUEABSOLUTE VALUE

2

CONTENT

2.1 Linear Equation

2.2 Quadratic Expression and Equations

2.3 Inequalities

2.4 Absolute value

2.1:2.1:

Linear Linear EquationsEquations

4

Objectives

• At the end of this topic, you should be able to• Define linear equations • Solve a linear equation• Solve equations that lead to linear equations • Solve applied problems involving linear

equations

5

Equation in one variable

A statements in which 2 expressions (sides) at least one containing the variable are equal

It may be TRUE or FALSE depending on the value of the variable.

The admissible values of the variable (those in the domain of the variable), if any, that result in a TRUE statement are called solutions or root.

To solve an equation means to find all the solutions of the equation

6

Equation in one variable, cont…

An equation will have only one solution or more than one solution or no real solutions or no solution

Solution set – the set of solutions of an equation, {a}

Identity – An equation that is satisfied for every value of the variable for which both sides are defined

Equivalent equations – Two or more equations that have the same solution set.

7

Linear Equations

A Linear Equation in one variable is equivalent to an equation of the form

where a and b are real numbers and

The linear equation has the single solution given by the formula

Simplify the given equations first, to solve a linear equations

0ax b 0a

bx

a

8

Steps for Solving a Linear Equation

STEP 1: If necessary, clear the equation of fractions by multiplying both sides by the least common multiple (LCM) of the denominators of all the fractions.

STEP 2: Remove all parentheses and simplify

STEP 3: Collect all terms containing the variable on one side and all remaining terms on the other side.

STEP 4: Check your solution (s)

9

Solve a Linear Equation

Solve the following equations

1. 3 4 2. 2 6 3

1 1 3 1 13. 5 4 2 1 4. 2

2 3 2 2 2

x x t t

x x y y

10

Solve equations that lead to linear equations


5 3 3 1 71. 3.

2 1 2 1 1 2

4 52. 5 4. 2 1 1 5 2 5

y 2

x x x x x x

y y y yy

11

An equation with no solution


3 31. 2

1 1

3 1 2 32.

2 2

x

x x

x x

x x

12

Translating Written/Verbal Information into a Mathematical Model

Addition Subtraction Multiplication Division Equals

And From Of Into Is

Plus Subtract Times Over Equals

More Less Product Divided by Same as

Added to Fewer By Quotient of Makes

Together with Minus Percent of Ratio of Leaves

Sum Difference Multiplied by a is to b Yields

Total Take away per Equivalent

Increased by Decreased by

Results in

13

Solve applied problems involving linear equations

Example 1 A total of Rp.45.000.000 is invested, some

in stocks and some in bonds. If the amount invested in bonds is half that invested in stocks, how much is invested in each category?

2.2:2.2:

Quadratic Quadratic Expression Expression &Equations&Equations

15

Objectives

• At the end of this topic you should be able to• Define quadratic expressions and equations

• Solve quadratic equations by factorization, square root method, and quadratic formula

• Recognize the types of roots of a quadratic equation based on the value of discriminant

• Solve applied problems involving quadratic equations

16

Quadratic Equations

A Quadratic Equation in is an equation equivalent to one of the form

where a, b and c are real numbers and

A Quadratic Equation in the form

is said to be in standard form

3 ways to solve quadratic equations a. Factoring

b. Square root method

c. Quadratic Formula

2 0ax bx c 0a

2 0ax bx c

17

Solve a Quadratic Equation by Factoring Solve the following equations

Repeated Solution / root of multiplicity 2 / double root When the left side, factors into 2 linear

equations with same solution

2 2

2 2

1. 5 6 0 3. 9 6 1 0

2. 2 3 4. 3 5 2 0

x x x x

x x x x

18

Solve a Quadratic Equation by the Square Root Method


221. 5 2. 2 16 x x

2If and 0, then x p p x p

19

Solve a Quadratic Equation by the Quadratic Formula Use the method of completing the square to obtain a

general formula for solving the quadratic equation


22 4

From 0 to 2

b b acax bx c x

a

2 2

2 2

1. 6 16 0 3. 2 8 5 0

2. 5 4 0 4. 2 3 24 0

x x x x

x x x x

20

Discriminant of a Quadratic Equation

For a Quadratic Equation

If there are two unequal real solutions

If there is a repeated solution, a root of

multiplicity 2

If there is no real solution (complex roots)

2 4 0b ac

2 0ax bx c

2 4 0b ac

2 4 0b ac

21

Examples

Find a real solutions, if any, of the following equations

2 2

22

2

1. 3 5 1 0 3. 3 4 2 0

25 3 22. 30 18 0 4. 9 0

2

5. 1 0 6. 1 7

x x x x

x xx x

x x x x

22

Application of Quadratic Equations

Example 1 The quadratic function

models the percentage of the U.S. population f (x), that was foreign-born x years after 1930. According to this model, in which year will 15% of the U.S. population be foreign-born?

20.0049 0.361 11.79f x x x

2.3:2.3:

Inequalities Inequalities

24

Objectives

• At the end of this topic you should be able to• Relate the properties of inequalities• Define and Solve linear inequalities• Define Solve quadratic inequalities• Understand and solve rational inequalities

involving linear and quadratic expression

[email protected] 25

Properties of Inequalities

1. If a 0 and b > 0 then a + b > 05. If a > 0 and b > 0 then ab > 06. If a 0 7. If a > b and –a < –b8. If a –b9. If a 0 then ac < bc 10. If a bc

11.

12. reciprocal property

13. reciprocal property

2 0

1If 0 then 0

1If 0 then 0

a

aa

aa


Solving Linear Inequalities

Solve the following inequality and graph the solution set

1

1. 3 2 5 4. 5 3 2 1

3 52. 4 7 2 3 5. 1 9

2

1 13. 9 5 2 1 6. 4 1 0

3 4

x x

xx x

x x x x

27

Solves problems involving linear inequalities

At least, minimum of, no less than

At most, maximum of, no more than

Is greater than, more than

Is less than, smaller than

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Examples

Sasha’s grade in her math course is calculated by the average of four tests. To receive an A for this course, she needs an average at least 89.5. If her current test scores are 84, 92, and 94, what range of scores can she make on the last test to receive an A for the course?

A painter charges RM80 plus RM1.50 per square foot. If a family is willing to spend no more than RM500, then what is the range of square footage they can afford?


Solving Quadratic inequalities

Step 1 - solve the related quadratic equation

Step 2 – plot the solution on a number line

Step 3 – Choose a test number from each interval & substitute the number into the inequality If the test number makes the inequality true

All numbers in that interval will solve the inequality If the test number makes the inequality false

No numbers in that interval will solve the inequality

Step 4 – State the solution set of the inequality ( It is a union of all intervals that solves the inequality) If the inequality symbols are or , then the values

from Step 2 are included. If the symbols are > or <, they are not solutions

30

Examples


22

22

1. 6 0 3. 1 2

2. 3 0 4. 1 2

x x x

x x x


Solving rational inequality

STEP 1: Solve the related equation

STEP 2: Find all values that make any denominator equal to 0

STEP 3: Plot the number found in Step 1 and 2 on a number line

STEP 4: Choose a test number from each interval and determine whether it solves the inequality.

STEP 5: The solution set is the union of all regions whose test number solves the inequality. If the inequality symbol is or , includes the values found in step 1

STEP 6: The solution set never includes the values found in Step 2 because they make the denominator equal to 0

32

Examples


4 1 21. 0 3.

2 1 1

5 22. 4 4. 3

1 4

x

x x x

x x

x x

2.4:2.4:

Absolute Value Absolute Value

34

Objectives

• At the end of this topic you should be able to• Define absolute value• Understand, state and use the properties of

absolute value• Solve problems on equations and inequalities

involving absolute value

35

What is Absolute Value

The absolute value can be define as:

, 0

, 0

a aa

a a

The absolute value represents the distance of a point on the number line from the origin

a

- a

36

Properties of Absolute Value

1. 0

2.

3.

4.

5.

6. , 0

a

a a

a b b a

a b b a

ab a b

aab

b b

For any real number a and b


Properties of Absolute Value

Equations involving absolute value

If is a positive real number and if is any algebraic expression, then

is equivalent to or

a y

y a y a y a

Inequalities involving absolute value

If a is positive real number and if is any algebraic expression, then

is equivalent to

is equivalent to

In other words, is equivalent to and

is equivalent to

a y

y a a y a

y a a y a

y a a y y a

y a y

and

is equivalent to and

a y a

u a u a u a

38

Solve equations involving absolute value

Solve the following equation

2

11. 4 13 3. 5 =1

2

2. 3 9 4. 16 =0

x x

x x

39

Solve inequalities involving absolute value

Solve the following inequalities. Graph the solution set

1. 2 4 3 3. 3

2. 1 4 <5 4. 2 5 3

x x

x x

40

Application of Absolute value

The inequality

describes the percentage of children in the population who think that being grounded is a bad thing about being kid. Solve the inequality and interpret the solution

9 2.9x

Thank YouThank You

EQUATIONS, INEQUALITIES & ABSOLUTE VALUE. 2 CONTENT 2.1 Linear Equation 2.2 Quadratic Expression and Equations 2.3 Inequalities 2.4 Absolute value.

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