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Inequalities
Prepared By:Malik Sabah-ud-din
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GENERAL OBJECTIVE
At the end of the chapter the students are expected to:
• Use interval notation.• Solve linear and nonlinear inequalities.• Solve application problems involving linear
inequalities.
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TODAY’S OBJECTIVE
• To identify an inequality.• To classify inequalities as absolute or conditional.• To use interval and set notation in writing solutions to inequalities.
• To represent graphically the solution to inequalities.• To apply intersection and union concepts in solving compound inequalities.
• To solve linear and fractional inequalities.• Understand that linear inequalities have one solution, no solution, or an interval solution.
At the end of the lesson the students are expected to:
DEFINITION
INEQUALITIESLet a and b denote two real numbers such that the graph of a on the number line is in the negative direction from the graph of b. Then we say that a is less than b and b is greater than a, or, in symbols:
A statement that one quantity is greater than or less than another quantity is called an INEQUALITY.
endpoints with esinequaliti are or esinequaliti strict called are or :Note
ab or ba
Absolute inequalities are inequalities which is true for all values of x.
Example:
KINDS OF INEQUALITIES
x1x
• Conditional inequalities are inequalities which is true for certain values of x.
Example: 51x
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FOUR WAYS OF EXPRESSING SOLUTIONS TO INEQUALITIES:
• inequality notation
• set notation
• interval notation
• graphical representation
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INEQUALITY
NOTATION
SET NOTATION
INTERVAL NOTATION
GRAPH/NUMBER LINE
bxa bxa|x b,a0 ba
[ )
0 ba οor
EXAMPLE
• a is the left endpoint • b is the right endpoint• If an inequality is a strict inequality (< or >)
parenthesis is used.• If an inequality includes an endpoint (> or <)
bracket is used.
INEQUALITY
NOTATION
SET NOTATION
INTERVAL NOTATION
GRAPH/NUMBER LINE
bxa bxax | ba,ba
[ ]
ba
or
Let x be a real number , x is ….
ax axx | a, a)
a
or
ax axx | a, b
a
]
ο
or
INEQUALITY
NOTATION
SET NOTATION
INTERVAL NOTATION
GRAPH/NUMBER LINE
bx bxx | ,bb(
b
or
Let x be a real number , x is ….
bx bxx |b[
b
or
,
,b
R R
ο
INEQUALITY
NOTATION
SET NOTATION
INTERVAL NOTATION
GRAPH/NUMBER LINE
bxa bxax | ba,0 ba
( )
0 baο
or
Let x be a real number , x is ….
bxa bxax | ba, 0 ba[ )
0 ba ο
or
ο
bxa bxax | ba, 0 ba(
0 ba
]
οor
Infinity is not a number. It is a symbol that means continuing indefinitely to the right on the number line.
Negative infinity means continuing indefinitely
to the left on the number line.
In interval notation, the lower number is always written on the left.
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Example 1
INEQUALITY NOTATION SET
NOTATIONGRAPHICAL REPRESENTATION
INTERVAL NOTATION
0 4-4
)
0 4-4○
(-∞,4)x < 4 4| xx
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INEQUALITY NOTATION
SET NOTATION
GRAPHICAL REPRESENTATION
INTERVAL NOTATION
x ≤ 4
0 4-4] 4| xx
0 4-4●
(-∞,4]
Example 2
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INEQUALITY NOTATION
SET NOTATION
GRAPHICAL REPRESENTATION
INTERVAL NOTATION
x > 4 4| xx
Example 3
0 4-4(
0 4-4○
(4, +∞)
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INEQUALITY NOTATION
SET NOTATION GRAPHICAL
REPRESENTATIONINTERVAL NOTATION
x ≥4 4| xx
Example 4
0 4-4
[
0 4-4●
[4, +∞)
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INEQUALITY
NOTATION
SET NOTATION
INTERVAL NOTATION
GRAPH/NUMBER LINE
4x1 4x1|x 4,14-1
[ )
4-1
οor
EXAMPLE 5
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INEQUALITY
NOTATION
SET NOTATION
INTERVAL NOTATION
GRAPH/NUMBER LINE
4x0 4x0|x 4,040
[ ]
40
or
EXAMPLE 5
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Example 6:
Classroom example 1.5.1 page 137
Express the following as an inequality and an interval. a. x is less than -1b. x is greater than or equal to 3c. x is greater than -2 and less than or equal to 7.
DEFINITION
UNION AND INTERSECTION
both or B or A in is x|x B A B. in elements the all with A in elements the all combining
by formed set the is B, A by denoted B, and A setsof union The
B and A in is x|x B A B. and A both in are that elements the by
formed set the is B, A by denoted B, and A setsof onintersecti The
A statement formed by joining two clauses with the word and is called a conjunction. For a conjunction to be true, both clauses must be true.
A statement formed by joining two clauses with the word or is called a disjunction. For a disjunction to be true, at least one of the clauses must be true.
DOUBLE OR COMBINED INEQUALITY
164X32 :Example
3x or 2x :Example
graph. and notation, interval in union and onintersecti the Express
D. C and D C find , 0,5 D and 3,3-CIf 138page
YourTurn
Example
notation. interval succinct more using 1,5-6,2- Express b.
notation. interval succinct more using 1,5-6,2- Express a. 138page
YourTurn
SOLVING LINEAR INEQUALITIES
SOLVING LINEAR INEQUALITIES
Linear inequalities are solved using the same procedure as linear equations with the following exception:
When you multiply or divide by a negative number, you must reverse the inequality sign.
Cross multiplication cannot be used with inequalities.
INEQUALITY PROPERTIES
PROCEDURES THAT DO NOT CHANGE THE INEQUALITY SIGN
x5 18x3
xx6 6x3
21 7x 29 8x7
3x 15x5
1. Simplifying by eliminating parentheses and collecting like terms.
2. Adding or subtracting the same quantity on both sides.
3.Multiplying or dividing by the same positive number.
INEQUALITY PROPERTIES
PROCEDURES THAT CHANGE (REVERSE) THE INEQUALITY SIGN
x4 to equivalent is 4x
1. Interchanging the two
sides of the inequality
2.Multiplying or dividing by the same negative number.
3x to equivalent is 15x5
233x-5 inequality the graph and Solve 139page
3 example .1
Example
SOLVING A LINEAR INEQUALITY
4-y35y2
notation interval in solution the express and Solve 143page
68# .2
5x23-4
notation interval in solution the express and Solve 143page
70# .3
Example
2x34
35x
inequality linear the Solve 140page
1.5.4 Ex. Classroom .1
SOLVING A LINEAR INEQUALITIES WITH FRACTION
y23y5
y521
y32
notation interval in solution the express and Solve 143page
75# .2
121
4s
33s
2s
notation interval in solution the express and Solve 143page
76# .3
Note: Common mistake is using cross multiplication to solve fractional inequalities.
Example
9x15- inequality linear the Solve 140page
1.5.5 Ex. Classroom .1
203
7x4- inequality linear the Solve
141page 1.5.6 Ex. Classroom
.2
SOLVING A DOUBLE OR COMPOUND LINEAR INEQUALITY
43
3y1
21
notation interval in solution the express and Solve 143page
85# .3
51
4z2
1-
notation interval in solution the express and Solve 143page
86# .4
SUMMARY
The solution to linear inequalities are solution sets that can be expressed in four ways:
1. Inequality notation2.Set Notation3.Interval Notation4.Graph (number line)
Linear inequalities are solved using the same procedures as linear equations with the following exception:
1. when you multiply or divide by a negative number you must reverse the inequality sign2. cross multiplication cannot be used with inequalities.
NON LINEAR INEQUALITIES IN ONE VARIABLE
TODAY’S OBJECTIVE
• To solve quadratic inequalities.• To solve polynomial inequalities.• To solve rational inequalities.• To solve absolute value inequalities• To solve application problems involving inequalities .
At the end of the lesson the students are expected to:
POLYNOMIAL INEQUALITIES
STEPS:
1. Write inequality in standard form (zero on one side).2. Identify zeros (factor if possible otherwise use quadratic formula)3. Draw the number line with zeros labeled.4. Determine the sign of the polynomial in each interval.5. Identify which interval(s) make the inequality true.6. Write the solution in interval notation.
Zeros of a polynomial are the values of x that make the polynomial equal to zero.These zeros divide the real number line into test intervals where the the value of the polynomial is either positive or negative.
SOLVING QUADRATIC INEQUALITY
Common mistakes:Taking the square root of both
sides.Dividing both sides by the
variable (x).
The square root method cannot be used for quadratic inequalities.
Dividing both sides by the variable (x) cannot be used for quadratic inequalities
SOLVING QUADRATIC INEQUALITY
04x4x 10.
04x4x 9.
150 page Turn Your 1x2x 8.
150 page #5 Example 1x2x 7.
149 page #4 Example x5x 6.
149 page #3 Example 3x2x 5.
148 page Ex.1.6.2 Classroom 151x 4.
147 page #2 Example 4x 3.
147 page Turn Your 6x5x .2
147 page #1 Example 12xx .1
2
2
2
2
2
2
2
2
2
2
Solve each quadratic inequality:
SOLVING A POLYNOMIAL INEQUALITY
02x5x21x 4.
04x5x 3.
150 page Turn Your 0x6xx 2.
150 page Ex.1.6.6 Classroom x3x25x 1.
2
24
23
32
Solve each inequality:
SOLVING A RATIONAL INEQUALITY
A rational expression have numerators and denominators , thus the
we have the following possible combinations:
, , ,
To solve rational inequalities we use a similar procedure for solving polynomial inequalities, with one exception. You must eliminate values for x that make the denominator equal to zero.
Once expressions are combined into a single fraction the value that make either the numerator or the denominator equal to zero divide the number line into intervals.
SOLVING A RATIONAL INEQUALITYSTEPS:
1. Write inequality in standard form (zero on one side).2. Identify zeros .
• Write as a single fraction• Determine values that make the numerator or denominator equal to zero
• Always exclude values that make the denominator = 0.
3. Draw the number line with zeros labeled.4. Determine the sign of the polynomial in each
interval.5. Identify which interval(s) make the inequality true.6. Write the solution in interval notation.
SOLVING A RATIONAL INEQUALITY
152 page #8 Example 32x
x 2.
151 page Ex.1.6.7 Classroom 0s-641s
1. 2
Solve each inequality:
154 page 57 # 4p
32p
12p
1 .4 2
154 page #43 p2p3
p42p-3p
.3 2
2
Week 5 Day 2
ABSOLUTE VALUE INEQUALITIES
PROPERTIES OF ABSOLUTE VALUE INEQUALITIES
ax or -ax to equivalent is ax 4.
ax or -ax to equivalent is ax 3.
axa- to equivalent is ax 2.
axa- to equivalent is ax .1
Week 5 Day 2
SOLVING AN ABSOLUTE VALUE INEQUALITY
Solve each inequality:
162 page 56 # 11x-4 .2
162 page #51 03x-4 .1
162 page 54 # 721x5 .3
162 page 58 # 52
5x32
.4
1x25x3 .5
4x32x5 .6
66x8x .7 2
Week 5 Day 2
Example
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APPLICATIONS INVOLVING LINEAR INEQUALITY
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course? the for B a least at earn to test fourth the on get can you score lowest the is What 84. and 77, 67, are acores test three first your class, biology general your In 3.
APPLICATIONS INVOLVING LINEAR INEQUALITY
SUMMARYThe following procedure can be used for solving polynomial and rational inequalities.
1. Write inequality in standard form (zero on one side).2. Determine the zeros; if it is a rational function, note
the domain restrictions. • Polynomial Inequality
- Factor if possible, otherwise, use quadratic formula
• Rational Inequality - Write as a single fraction - Determine values that make the numerator
or denominator equal to zero -Always exclude values that make the
denominator = 0. 3. Draw the number line with zeros labeled.4. Determine the sign of the polynomial in each
interval.5. Identify which interval(s) make the inequality true.6. Write the solution in interval notation.