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In this section you will:
  • Use interval notation.
  • Use properties of inequalities.
  • Solve inequalities in one variable algebraically.
  • Solve absolute value inequalities.
Several red winner’s ribbons lie on a white table.

It is not easy to make the honor role at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.

Using interval notation

Indicating the solution to an inequality such as x 4 can be achieved in several ways.

We can use a number line as shown in [link] . The blue ray begins at x = 4 and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.

A number line starting at zero with the last tick mark being labeled 11.  There is a dot at the number 4 and an arrow extends toward the right.

We can use set-builder notation : { x | x 4 } , which translates to “all real numbers x such that x is greater than or equal to 4.” Notice that braces are used to indicate a set.

The third method is interval notation    , in which solution sets are indicated with parentheses or brackets. The solutions to x 4 are represented as [ 4 , ) . This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.

The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval    , or a set of numbers in which a solution falls, are [ −2 , 6 ) , or all numbers between −2 and 6 , including −2 , but not including 6 ; ( 1 , 0 ) , all real numbers between, but not including −1 and 0 ; and ( , 1 ] , all real numbers less than and including 1. [link] outlines the possibilities.

Set Indicated Set-Builder Notation Interval Notation
All real numbers between a and b , but not including a or b { x | a < x < b } ( a , b )
All real numbers greater than a , but not including a { x | x > a } ( a , )
All real numbers less than b , but not including b { x | x < b } ( , b )
All real numbers greater than a , including a { x | x a } [ a , )
All real numbers less than b , including b { x | x b } ( , b ]
All real numbers between a and b , including a { x | a x < b } [ a , b )
All real numbers between a and b , including b { x | a < x b } ( a , b ]
All real numbers between a and b , including a and b { x | a x b } [ a , b ]
All real numbers less than a or greater than b { x | x < a and x > b } ( , a ) ( b , )
All real numbers { x | x  is all real numbers } ( , )

Using interval notation to express all real numbers greater than or equal to a

Use interval notation to indicate all real numbers greater than or equal to −2.

Use a bracket on the left of −2 and parentheses after infinity: [ −2 , ) . The bracket indicates that −2 is included in the set with all real numbers greater than −2 to infinity.

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Use interval notation to indicate all real numbers between and including −3 and 5.

[ −3 , 5 ]

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Questions & Answers

How look for the general solution of a trig function
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Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
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simplify each radical by removing as many factors as possible (a) √75
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what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
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litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
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Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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