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Inequalities Inequalities

In this lesson, you are expected to: In this lesson, you are expected to:

1.1. Differentiate between equations and inequalities,Differentiate between equations and inequalities,

2.2. Find the solution of an inequality involving one Find the solution of an inequality involving one variable.variable.

Differences between Linear Equation and LinearDifferences between Linear Equation and Linear

Inequalities:Inequalities:

InequalityInequality – – A mathematical statement where one algebraic expression is not equal to another A mathematical statement where one algebraic expression is not equal to another algebraic expression.algebraic expression.

x + 5 ≤ 6x + 5 ≤ 6

7 (x + 2) < - 57 (x + 2) < - 5

5 > 2x + 45 > 2x + 4

4 ≠ 5x + 24 ≠ 5x + 2

x + 2 ≥ 7x + 2 ≥ 7

a.a. The values of variable x arThe values of variable x are set of all numbers greater that e set of all numbers greater that -4 going to positive infinity but not including -4.-4 going to positive infinity but not including -4.

b.b. The values of variable x arThe values of variable x are set are set of all numbers in e set are set of all numbers in the number line not including +8.the number line not including +8.

c.c. The values of variable x arThe values of variable x are set of all number from + 9 e set of all number from + 9 to positive infinity.to positive infinity.

d.d. The values of variable x are set of all number from -The values of variable x are set of all number from -

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to negative infinity. to negative infinity.

e.e. The values of variable x arThe values of variable x are set of all numbers between -5 e set of all numbers between -5 to +5.to +5.

f.f. The The values of variable x are sevalues of variable x are set of all numbers frot of all numbers from 0 up to the number m 0 up to the number before 5.before 5.

1.1. x > 7x > 7

2.2. x < 14x < 14

3.3. --4 ≥ 54 ≥ 5

4.4. 10 ≤ 2510 ≤ 25

5.5. x x ≠≠ -12-12

Symbols =Symbols = ≠ , <, >, ≤, ≥≠ , <, >, ≤, ≥

Number Number of of Solution Solution Exactly Exactly one one / / one one / / infinite infinite More More than than oneone

Graph Graph of of Solution Solution A A point point Set Set of of pointspoints

Example: Example: x x + + 7 7 = = 25 25 3x 3x + + 4 4 > > 1616

≠≠ Not Not equalequal

< < Less Less thanthan

> > Greater Greater thanthan

≤≤ Less Less than than or or equal equal toto

≥≥ Greater Greater than than or or equal equal toto

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

Operations on integers, Order of real number Operations on integers, Order of real number

x > -4x > -4

x ≠ 8x ≠ 8

x ≥ 9x ≥ 9

xx ≤≤

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-3 < x < 3-3 < x < 3

0 ≤ x < 50 ≤ x < 5

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are real numbers. If , and

-3 < x ≤ 1

and and

>

and

and

and

2. x - 8 < 11

Solution: x – 8 < - 11

x – 8

x < -3

1. x + 7 ≥ 0

Solution: x + 7 ≥ 0

x + 7

x ≥ - 7

3. Given: 5x – 9 > -2x + 5

Solution: 5x – 9

5x > -2x + 14

5x

(

)7x > 14(

)

x > 2

4. Given: 2(x - 3) ≤ 5x +12

Solution: 2x - 6 ≤ 5x + 12

2x ≤ 5x + 18

2x + 18

(-

) -3x ≤ 18 (-

)

x ≥ - 6

≥ to ≤ and vice versa,

> to < and vice versa.

0 5 10-5-10

0 5 10-5-10

Given: -3< x + 2 < 14

Solution: -3

-5 < x < 12

Given: -1 ≤ 2 – 3x < 11

Solution: -1

-3 < 9

1 ≥ x >

0 10 15-5-10

0 5 10-5-10

let a, b, and c , then .

let a, b, and c are real numbers and c≠ 0.

x > 2 :

x ≥ - 6:

-5 < x < 12:

1 ≥ x > -3:

and

>

and

<

and

>

< - 11 ≥ 0

> -2x + 5

> -2x + 14 ≤ 5x

< x + 2 < 14 ≤ 2 – 3x < 11

≤ – 3x

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1. x + 4 7 =

2. x - 5 =

3. x + 5 =

4. x – 6 =

5. x + 8 =

6. x – 9 =

7. 2x =

8. 3x =

9. -5x 25 =

10. -4x 24 =

1. x + 47 =

2. y - 83 =

3. -2 + 9x 10x =

4. 14 - 18y =

5. 15x - 4 =

6. 13 - y =

7. 5x + 7 =

8. -23 3x – 11 =

9. -7 x + 14 8 =

10. -1 + 2y =

14

12

11

-4

-13

18

-9

36

-79

-7y

11x – 16

29 + 2y

3(x + 1)

-2

5 - 2y

Inequalities Inequalities

In this lesson, you are expected to: In this lesson, you are expected to:

1.1. Differentiate between equations and inequalities,Differentiate between equations and inequalities,

2.2. Find the solution of an inequality involving one Find the solution of an inequality involving one variable.variable.

Differences between Linear Equation and LinearDifferences between Linear Equation and Linear

Inequalities:Inequalities:

InequalityInequality – – A mathematical statement where one algebraic expression is not equal to another A mathematical statement where one algebraic expression is not equal to another algebraic expression.algebraic expression.

x + 5 ≤ 6x + 5 ≤ 6

7 (x + 2) < - 57 (x + 2) < - 5

5 > 2x + 45 > 2x + 4

4 ≠ 5x + 24 ≠ 5x + 2

x + 2 ≥ 7x + 2 ≥ 7

a.a. The values of variable x arThe values of variable x are set of all numbers greater that e set of all numbers greater that -4 going to positive infinity but not including -4.-4 going to positive infinity but not including -4.

b.b. The values of variable x arThe values of variable x are set are set of all numbers in e set are set of all numbers in the number line not including +8.the number line not including +8.

c.c. The values of variable x arThe values of variable x are set of all number from + 9 e set of all number from + 9 to positive infinity.to positive infinity.

d.d. The values of variable x are set of all number from -The values of variable x are set of all number from -

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to negative infinity. to negative infinity.

e.e. The values of variable x arThe values of variable x are set of all numbers between -5 e set of all numbers between -5 to +5.to +5.

f.f. The The values of variable x are sevalues of variable x are set of all numbers frot of all numbers from 0 up to the number m 0 up to the number before 5.before 5.

1.1. x > 7x > 7

2.2. x < 14x < 14

3.3. --4 ≥ 54 ≥ 5

4.4. 10 ≤ 2510 ≤ 25

5.5. x x ≠≠ -12-12

Symbols =Symbols = ≠ , <, >, ≤, ≥≠ , <, >, ≤, ≥

Number Number of of Solution Solution Exactly Exactly one one / / one one / / infinite infinite More More than than oneone

Graph Graph of of Solution Solution A A point point Set Set of of pointspoints

Example: Example: x x + + 7 7 = = 25 25 3x 3x + + 4 4 > > 1616

≠≠ Not Not equalequal

< < Less Less thanthan

> > Greater Greater thanthan

≤≤ Less Less than than or or equal equal toto

≥≥ Greater Greater than than or or equal equal toto

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

0 0 55 1010-5-5-10-10

Operations on integers, Order of real number Operations on integers, Order of real number

x > -4x > -4

x ≠ 8x ≠ 8

x ≥ 9x ≥ 9

xx ≤≤

22

33

-3 < x < 3-3 < x < 3

0 ≤ x < 50 ≤ x < 5

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are real numbers. If , and

-3 < x ≤ 1

and and

>

and

and

and

2. x - 8 < 11

Solution: x – 8 < - 11

x – 8

x < -3

1. x + 7 ≥ 0

Solution: x + 7 ≥ 0

x + 7

x ≥ - 7

3. Given: 5x – 9 > -2x + 5

Solution: 5x – 9

5x > -2x + 14

5x

(

)7x > 14(

)

x > 2

4. Given: 2(x - 3) ≤ 5x +12

Solution: 2x - 6 ≤ 5x + 12

2x ≤ 5x + 18

2x + 18

(-

) -3x ≤ 18 (-

)

x ≥ - 6

≥ to ≤ and vice versa,

> to < and vice versa.

0 5 10-5-10

0 5 10-5-10

Given: -3< x + 2 < 14

Solution: -3

-5 < x < 12

Given: -1 ≤ 2 – 3x < 11

Solution: -1

-3 < 9

1 ≥ x >

0 10 15-5-10

0 5 10-5-10

let a, b, and c , then .

let a, b, and c are real numbers and c≠ 0.

x > 2 :

x ≥ - 6:

-5 < x < 12:

1 ≥ x > -3:

and

>

and

<

and

>

< - 11 ≥ 0

> -2x + 5

> -2x + 14 ≤ 5x

< x + 2 < 14 ≤ 2 – 3x < 11

≤ – 3x

Page 3

1. x + 4 7 =

2. x - 5 =

3. x + 5 =

4. x – 6 =

5. x + 8 =

6. x – 9 =

7. 2x =

8. 3x =

9. -5x 25 =

10. -4x 24 =

1. x + 47 =

2. y - 83 =

3. -2 + 9x 10x =

4. 14 - 18y =

5. 15x - 4 =

6. 13 - y =

7. 5x + 7 =

8. -23 3x – 11 =

9. -7 x + 14 8 =

10. -1 + 2y =

14

12

11

-4

-13

18

-9

36

-79

-7y

11x – 16

29 + 2y

3(x + 1)

-2

5 - 2y