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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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.