Transcript EQUATIONS, INEQUALITIES & ABSOLUTE VALUE

EQUATIONS,
INEQUALITIES &
ABSOLUTE VALUE
CONTENT
2.1 Linear Equation
2.2 Quadratic Expression and Equations
2.3 Inequalities
2.4 Absolute value
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2.1:
Linear
Equations
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
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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
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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.
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Linear Equations
 A Linear Equation in one variable is equivalent to
an equation of the form
ax  b  0
where a and b are real numbers and a  0
 The linear equation has the single solution given
by the formula
x
b
a
 Simplify the given equations first, to solve a
linear equations
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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)
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Solve a Linear Equation
 Solve the following equations
1. 3 x  4  x
2. 2t  6  3  t
1
1
3.
 x  5  4   2 x  1
2
3
3
1 1
4.
y2  y
2
2 2
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Solve equations that lead to linear
equations
 Solve the following equations
1.
5
3

x2
x 1
4
5
2.  5 
y
2y
3.
3
1
7


x2
x  1  x  1 x  2 
4.
 2 y  1 y  1   y  5  2 y  5 
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An equation with no solution
 Solve the following equations
1.
3x
3
2
x 1
x 1
3x  1 2  3x
2.

x2
2 x
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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
Increased by Decreased by
Equivalent
Results in
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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?
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2.2:
Quadratic
Expression
&Equations
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
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Quadratic Equations
 A Quadratic Equation in is an equation
equivalent to one of the form
ax2  bx  c  0
where a, b and c are real numbers and a  0
 A Quadratic Equation in the form
ax2  bx  c  0
is said to be in standard form
 3 ways to solve quadratic equations
a. Factoring
b. Square root method
c. Quadratic Formula
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Solve a Quadratic Equation by
Factoring
 Solve the following equations
1. x 2  5 x  6  0
3. 9 x 2  6 x  1  0
2. 2x 2  x  3
4. 3x 2  5 x  2  0
 Repeated Solution / root of multiplicity 2 /
double root

When the left side, factors into 2 linear
equations with same solution
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Solve a Quadratic Equation by the
Square Root Method
If x  p and p  0, then x   p
2
 Solve the following equations
1. x  5
2
2.
 x  2
2
 16
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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
From ax 2  bx  c  0
to
b  b2  4ac
x
2a
 Solve the following equations
1. x 2  6 x  16  0
2.
x2  5x  4  0
3. 2 x 2  8 x  5  0
4. 2x 2  3 x  24  0
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Discriminant of a Quadratic Equation
For a Quadratic Equation

b2  4ac  0
If


there are two unequal real solutions
b2  4ac  0
If


ax2  bx  c  0
there is a repeated solution, a root of
multiplicity 2
b2  4ac  0
If

there is no real solution (complex roots)
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Examples
 Find a real solutions, if any, of the following
equations
1. 3 x 2  5 x  1  0
3. 3 x 2  4 x  2  0
2.
25 2
x  30 x  18  0
2
3 2
4. 9   2  0
x x
5.
x2  x  1  0
6.
x 1  x  7
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Application of Quadratic Equations
 Example 1

2
f
x

0.0049
x
 0.361x 11.79


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?
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2.3:
Inequalities
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
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Properties of Inequalities
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
If a < b and b < c then a < c
If a < b and c is any number, then a + c < b + c
If a < b and c is any number, then a – c < b – c
If a > 0 and b > 0 then a + b > 0
If a > 0 and b > 0 then ab > 0
If a < b then b – a > 0
If a > b and –a < –b
If a < b and –a > –b
If a < b and c > 0 then ac < bc
If a < b and c < 0 then ac > bc
a2  0
1
0
12.
a
13. If a  0 then 1  0
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a
If a  0 then
reciprocal property
reciprocal property
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Solving Linear Inequalities
 Solve the following inequality and graph the
solution set
1. 3  2 x  5
2.
4x  7  2x  3
3.
1
1
 9 x  5  2 x  x  1
3
4
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4.  5  3 x  2  1
3  5x
5.  1 
9
2
6.
 4 x  1
1
0
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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?
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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.
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 If the symbols are > or <, they are not solutions
Examples
 Solve the following inequality and graph the
solution set
1. x  x  6  0
2
2. x  3 x  0
2
3.
 x  1
4.
 x 1 
2
2
 2
 2
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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
includes the values found in step 1
 or  ,
 STEP 6: The solution set never includes the values found in
Step 2 because they make the denominator equal to 0
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Examples
 Solve the following inequality and graph the
solution set
1.
2.
x4
0
x2
x5
4
x 1
3.
1
2

x 1
x 1
4.
x2
3
x4
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2.4:
Absolute Value
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
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What is Absolute Value
 The absolute value can be define as:
a, a  0
a 
a, a  0
 The absolute value represents the distance of a point on the
number line from the origin
a
-a
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Properties of Absolute Value
 For any real number a and b
1.
a 0
2.
a  a
3.
ab  ba
4.
a b  ba
5.
ab  a b
6.
a
a

b
b
,b 0
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Properties of Absolute Value
 Equations involving absolute value
If a is a positive real number and if y is any algebraic expression, then
y  a is equivalent to y  a or y  a
 Inequalities involving absolute value
If a is a positive real number and if y is any algebraic expression, then
y  a is equivalent to  a  y  a
y  a is equivalent to  a  y  a
In other words, y  a is equivalent to  a  y and y  a
y  a is equivalent to y  a and y  a
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u  a is equivalent to u  a and u  a
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Solve equations involving absolute
value
 Solve the following equation
1.
x  4  13
1
3. 5  x =1
2
2.
3 x9
4. x  16 =0
2
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Solve inequalities involving absolute
value
 Solve the following inequalities. Graph the
solution set
1.
2x  4  3
3.
x 3
2.
1  4 x <5
4. 2 x  5  3
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Application of Absolute value
 The inequality
x  9  2.9
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
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Thank You