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Inequalities - Interval Notation










































how do you put this in interval notation? 2x^2+x greater or equal to 1

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Jan 19, 2012
Inequalities - Interval Notation
by: Staff

Question:


how do you put this in interval notation?
2x^2+x greater or equal to 1


Answer:

2x² + x ≥ 1

Subtract 1 from each side of the equation

2x² + x - 1 ≥ 1 - 1

2x² + x - 1 ≥ 0


Factor the equation

(2x - 1)(x + 1) ≥ 0


Find the two solutions of the equation

1st root

2x - 1 = 0

2x - 1 + 1 = 0 + 1

2x - 1 + 1 = 1

2x + 0 = 1

2x = 1

2x/2 = ½

x*(2/2) = ½

x*(1) = ½

x = ½


2nd root

x + 1 = 0

x + 1 - 1 = 0 - 1

x + 0 = 0 - 1

x = 0 - 1

x = - 1


the final solution of valid x values is:

x ≤ -1 or x ≥ ½


the final solution (set Z) of valid x values in SET BUILDER NOTATION is:

Z = {x | x ∈ ℝ, x ≤ -1 or x ≥ ½}

{} curly brackets surround the expression, indicating “this is a set”

∈ = element of a set

| and : can be used interchangeably. Both notations are separators which mean “where” or “such that”

x: the first “x” is the “output function”, shown as = {x |

x: the second “x” is the “variable”, shown as | x ∈ ℝ,

ℝ (the set of all real numbers) is the“input set”

x ≤ -1 or x ≥ ½ is the “predicate”


Reading from left to right: “Z” is the set of all numbers “x” {x | … , …} where “x” is an element of the set of real numbers ℝ { … | x ∈ ℝ, … } and “x” is less than or equal to -1 OR greater than or equal to ½ {… | … , x ≤ -1 or x ≥ ½ }.


>>> the final solution of valid x values in INTERVAL NOTATION is:

x ∈ (-∞,-1] ∪ [0.5,∞)

Interval notations:

∈ = element of a set

∪ = union

Both “(” and “)” = open, value NOT INCLUDED

Both “[” and “]” = closed, value INCLUDED

[0.5,∞) = half open interval, left value of 0.5 INCLUDED, right value of ∞ NOT INCLUDED

(-∞,-1] = half open interval, left value of -∞ NOT INCLUDED, right value of -1 INCLUDED

To view a graph of the solution in interval notation, open the link shown below.

(1) If your browser is Firefox, click the following link to VIEW the solution; or if your browser is Chrome, Internet Explorer, Opera, or Safari (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

(Use the Backspace key to return to this page.)

http://www.solving-math-problems.com/images/Inequalities-Interval-Notation-graph-01-2012-01-19.png



Thanks for writing.
Staff

www.solving-math-problems.com



Jun 02, 2014
Inequalities - Interval Notation
by: Anonymous

how would you put in 4y-6 is less than 18 or 2y-2 is greater than -8

Jun 04, 2014
Inequalities - Interval Notation
by: Staff



Part A

4y - 6 is less than 18

4y - 6 < 18

Add 6 to each side of the equation

4y – 6 + 6 < 18 + 6

4y + 0 < 18 + 6

4y < 24

Divide each side of the equation by 4

4y / 4 < 24 / 4


y * (4 / 4) < 6 * (4 / 4)

y * 1 < 6 * 1

y < 6


the final solution of valid y values is:

y < 6


>>> the final solution of valid y values in INTERVAL NOTATION is:

y ∈ (-∞, 6)

Interval notations:
∈ = element of a set
Both “(” and “)” = open, value NOT INCLUDED



 interval notation 4y - 6 <  18 

*** Click to enlarge image ***






----------------------------

Jun 04, 2014
Inequalities - Interval Notation
by: Staff



----------------------------


Part B

2y - 2 is greater than -8

2y - 2 > -8

Add 2 to each side of the equation

2y - 2 + 2 > -8 + 2

2y + 0 > -8 + 2

2y > -6

Divide each side of the equation by 2

2y / 2 > -6 / 2

y * (2 / 2) > (-3) * (2 / 2)

y * 1 > (-3) * 1

y > -3

the final solution of valid y values is:

y > -3


>>> the final solution of valid y values in INTERVAL NOTATION is:

y ∈ (-3, ∞)

Interval notations:
∈ = element of a set
Both “(” and “)” = open, value NOT INCLUDED



interval notation 2y - 2  >  -8

*** Click to enlarge image ***









Thanks for writing.
Staff

www.solving-math-problems.com


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