3 1 the real line and linear inequalities-x

Inequalities

We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.

Inequalities


-2 20 1 3 +-1-3–

Inequalities


-2 20 1 3 +-1-3–

2/3

Inequalities


-2 20 1 3 +-1-3–

2/3 2½

Inequalities


-2 20 1 3 +-1-3–

2/3 2½ π 3.14..

Inequalities

–π –3.14..


-2 20 1 3 +-1-3–

2/3 2½ π 3.14.. This line with each position addressed by a real number is called the real (number) line.

Inequalities

–π –3.14..


-2 20 1 3 +-1-3–


Inequalities

–π –3.14..

Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.


-2 20 1 3 +-1-3–


Inequalities

+–RL

–π –3.14..



-2 20 1 3 +-1-3–


Inequalities

+–R

We write this as L < R and called this the natural form because it corresponds to their respective positions on the real line.

L <

–π –3.14..



-2 20 1 3 +-1-3–


Inequalities

+–R

We write this as L < R and called this the natural form because it corresponds to their respective positions on the real line. This relation may also be written as R > L (less preferable).

L <

–π –3.14..


Example A. 2 < 4, –3< –2, 0 > –1 are true statementsInequalities

Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.

Inequalities


Inequalities

If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x".


Inequalities

If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a).


Inequalities

If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,

+–a

open dot

a < x


Inequalities


+–a

open dot

If we want all the numbers x greater than or equal to a (including a), we write it as a < x.

a < x


Inequalities


+–a

open dot

If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture

+–a

solid dot

a < x

a < x


Inequalities


+–a

open dot


+–a

solid dot

a < x

a < x

The numbers x fit the description a < x < b where a < b are all the numbers x between a and b.


Inequalities


+–a

open dot


+–a

solid dot

a < x

a < x

The numbers x fit the description a < x < b where a < b are all the numbers x between a and b.

+–a a < x < b b


Inequalities


+–a

open dot


+–a

solid dot

a < x

a < x

The numbers x fit the description a < x < b where a < b are all the numbers x between a and b. A line segment as such is called an interval.

+–a a < x < b b

To solve inequalities:Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.

To solve inequalities:1. Simplify both sides of the inequalities

Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.


Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side


Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x


Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18

To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule).

Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18


Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)


Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x


Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x

To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.

Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x


Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x–12/3 3x/3

>

div. by 3 (no need to switch >)


Inequalities

Example B. Solve 3(2 – x) > 2(x + 9) – 2x. Draw the solution.3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x–12/3 3x/3

>

–4 > x or x < –4

div. by 3 (no need to switch >)

+–4

Example C. (Interval Inequality)Solve 12 > –2x + 6 > –4 and draw

InequalitiesWe also have inequalities in the form of intervals.

Example C. (Interval Inequality)Solve 12 > –2x + 6 > –4 and draw.

InequalitiesWe also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities.

Example C. (Interval Inequality)Solve 12 > –2x + 6 > –4 and draw.

InequalitiesWe also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, then divide or multiply to get x.

Example C. (Interval Inequality)Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 subtract 6 –6 –6 –6


Example C. (Interval Inequality)Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10


Example C. (Interval Inequality)Solve 12 > –2x + 6 > –4 and draw. 12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10 div. by -2, switch inequality sign 6 -2

-2x -2< -10

-2<



-3 < x < 5

div. by -2, switch inequality sign 6 -2

-2x -2< -10

-2<



-3 < x < 5


-2x -2< -10

-2<

InequalitiesWe also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, then divide or multiply to get x. The answer is an interval of numbers.


0+

-3 < x < 5

5


-2x -2< -10

-2<

-3

InequalitiesWe also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, then divide or multiply to get x. The answer is an interval of numbers.

Comparison Statements, Inequalities and Intervals

The following adjectives or comparison phrases are translated into inequalities in mathematics:


The following adjectives or comparison phrases are translated into inequalities in mathematics:“positive” vs. “negative”,“non–positive” vs. ”non–negative”,“more/greater than” vs. “less/smaller than”,“no more/greater than” vs. “no less/smaller than”, “at least” vs. ”at most”,



“Positive” vs. “Negative”


0


A quantity x is positive means that 0 < x,

0




A quantity x is positive means that 0 < x,

0+ x is positive




A quantity x is positive means that 0 < x, and that x is negative means that x < 0.

0+ x is positive





0+– x is negative x is positive





0+– x is negative x is positive



The phrase “the temperature T is positive” is “0 < T”.

“Non–Positive” vs. “Non–Negative”Comparison Statements, Inequalities and Intervals

A quantity x is non–positive means x is not positive, or “x ≤ 0”,


A quantity x is non–positive means x is not positive, or “x ≤ 0”,

“Non–Positive” vs. “Non–Negative”

0+–

x is non–positive


A quantity x is non–positive means x is not positive, or “x ≤ 0”, and that x is non–negative means x is not negative, or “0 ≤ x”.

0+–

x is non–positive

x is non–negative



0+–

x is non–positive

x is non–negative

The phrase “the account balance A is non–negative” is “0 ≤ A”.



0+–

x is non–positive

x is non–negative



“More/greater than” vs “Less/smaller than”



0+–

x is non–positive

x is non–negative



“More/greater than” vs “Less/smaller than”Let C be a number, x is greater than C means “C < x”,

C

x is more than C



0+–

x is non–positive

x is non–negative



“More/greater than” vs “Less/smaller than”Let C be a number, x is greater than C means “C < x”, and that x is less than C means “x < C”.

C x is less than C

x is more than C


“No more/greater than” vs “No less/smaller than”and “At most” vs “At least”


“No more/greater than” vs “No less/smaller than”

A quantity x is “no more/greater than C” is the same as “x is at most C” and means “x ≤ C”.

and “At most” vs “At least”



A quantity x is “no more/greater than C” is the same as “x is at most C” and means “x ≤ C”.

+– x is no more than C C


x is at most C



A quantity x is “no more/greater than C” is the same as “x is at most C” and means “x ≤ C”. A quantity x is “no–less than C” is the same as “x is at least C” and means “C ≤ x”.

+– x is no more than C x is no less than CC


x is at most C x is at least C




+– x is no more than C x is no less than C

“The account balance A is no–less than 500” is the same as “A is at least 500” or that “500 ≤ A”.

C






+– x is no more than C x is no less than C

“The temperature T is no–more than 250o” is the same as “T is at most 250o” or that “T ≤ 250o”.

“The account balance A is no–less than 500” is the same as “A is at least 500” or that “500 ≤ A”.

C




We also have the compound statements such as “x is more than a, but no more than b”.


We also have the compound statements such as “x is more than a, but no more than b”. In inequality notation, this is “a < x ≤ b”.



+– a a < x ≤ b b



+– a a < x ≤ b b

and it’s denoted as: (a, b] where “(”, “)” means the end points are excluded and that “[”, “]” means the end points are included.



+– a a < x ≤ b b

and it’s denoted as: (a, b] where “(”, “)” means the end points are excluded and that “[”, “]” means the end points are included.A line segment as such is called an interval.


Therefore the statement “the length L of the stick must be more than 5 feet but no more than 7 feet” is “5 < L ≤ 7”


+– a a < x ≤ b b



Therefore the statement “the length L of the stick must be more than 5 feet but no more than 7 feet” is “5 < L ≤ 7” or that L must be in the interval (5, 7].


+– a a < x ≤ b b


75 5 < L ≤ 7 or (5, 7]

L


Therefore the statement “the length L of the stick must be more than 5 feet but no more than 7 feet” is “5 < L ≤ 7” or that L must be in the interval (5, 7].


+– a a < x ≤ b b


75 5 < L ≤ 7 or (5, 7] Following is a list of interval notation.

L


Let a, b be two numbers such that a < b, we writeba


Let a, b be two numbers such that a < b, we write

or a ≤ x ≤ b as [a, b],ba




or a < x < b as (a, b),ba






Note: The notation “(2, 3)” is to be viewed as an interval or as a point (x, y) depends on the context.




or a ≤ x < b as [a, b),

ba

or a < x ≤ b as (a, b],

ba


Using the “∞” symbol which means to “surpass all finite numbers”, we may write the rays

∞a or a ≤ x, as [a, ∞),




or a ≤ x < b as [a, b),

ba

or a < x ≤ b as (a, b],

ba



∞a or a ≤ x, as [a, ∞),

∞a or a < x, as (a, ∞),




or a ≤ x < b as [a, b),

ba

or a < x ≤ b as (a, b],

ba



∞a or a ≤ x, as [a, ∞),

–∞ aor x ≤ a, as (–∞, a],

∞a or a < x, as (a, ∞),

–∞ a or x < a, as (–∞, a),




or a ≤ x < b as [a, b),

ba

or a < x ≤ b as (a, b],

ba


InequalitiesExercise. A. Draw the following Inequalities. Indicate clearly whether the end points are included or not.1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12 B. Write in the natural form then draw them.5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12 C. Draw the following intervals, state so if it is impossible.9. 6 > x ≥ 3 10. –5 < x ≤ 2 11. 1 > x ≥ –8 12. 4 < x ≤ 213. 6 > x ≥ 8

14. –5 > x ≤ 2 15. –7 ≤ x ≤ –3 16. –7 ≤ x ≤ –9D. Solve the following Inequalities and draw the solution.17. x + 5 < 3

18. –5 ≤ 2x + 3 19. 3x + 1 < –8 20. 2x + 3 ≤ 12 – x 21. –3x + 5 ≥ 1 – 4x

22. 2(x + 2) ≥ 5 – (x – 1) 23. 3(x – 1) + 2 ≤ – 2x – 924. –2(x – 3) > 2(–x – 1) + 3x 25. –(x + 4) – 2 ≤ 4(x – 1)26. x + 2(x – 3) < 2(x – 1) – 227. –2(x – 3) + 3 ≥ 2(x – 1) + 3x + 13

Inequalities

F. Solve the following interval inequalities.

28. –4 ≤ 2x 29. 7 > 3

–x 30. < –4–xE. Clear the denominator first then solve and draw the solution.

5x 2 3

1 23 2 + ≥ x31. x 4

–3 3

–4 – 1 > x32.

x 2 83 3

45 – ≤ 33. x 8 12

–5 7 1 + > 34.

x 2 3–3 2

3 4

41 – + x35. x 4 6

5 53

–1 – 2 + < x36.

x 12 27 3

6 1

43 – – ≥ x37.

≤

40. – 2 < x + 2 < 5 41. –1 ≥ 2x – 3 ≥ –1142. –5 ≤ 1 – 3x < 10 43. 11 > –1 – 3x > –7

38. –6 ≤ 3x < 12 39. 8 > –2x > –4

3 1 the real line and linear inequalities-x

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