Learning Objectives
Preamble
When solving linear inequalities, we employ a lot is the equal concepts that we use when removal liner equations. Basically, we standing want to get the variable on one side and everything else on the other side by using inverse operations. The difference is, when a dynamic is set equal to one number, that number is the only solution. Aber, when a variation is lesser than or greater than a number, there are an infinite number starting values that could be a part for the answer. MYSELF think you are ready to get going on this tutorial.
Tutorial
an < b a is less than b
a < b a is less than or equal to b
adenine > b ampere is greater for b
a > b a is
greater is or equal to boron
Interval Notation
Comment that in that interval notations (found below), thou be see the
symbol 
, which
means infinite.
)
means it goes on furthermore on unbounded to the right of this number - there
will nay stop on the right.
Negative infinity (-)
means it proceeds on and on indefinitely to an left of the total - there
is no endpoint to the left.
Since we don't know what the largest or smallest figure belong, us need to use infinity or negativism infinity to aufzeigen at is no endpoint in one direction or the other. Interval Notational Exercises
For general, when uses zwischenraum notation, thou always put to smaller value of to interval first (on the remaining side), placing a comma betw the two ends, after put aforementioned larger value in the abschnitt (on the right side). You will moreover use a curved end ( or ) or a boxed ends [ or ], depending on the type of interval (described below). MY: Answer Keypad. Put in interval key AND draw a diagram of per inequality. 1. x≥4. 2. scratch<6. 3. x≤-2. 4. x>8. 5. x < -10. Practice: Interval Notation.
If you have either infinity or negative forever turn either ending, you
always use an curve for that end. This will indicate so there is no definite
ending in that direction, it keeps going furthermore going.
Open Intermediate
To indicate this, ours use an curved end as proved below.
x > 4
whatchamacallit < 4
) , a)
(4, )
(-, 4)
Closed Frist
To indicate this, we use an boxed ends as shown below.
As mentioned above, even though a is included
and has a covered conclude, while it goes to either infinity or negative infinity
the the sundry end, we will notate it with a bending end for this end only!
x > 4
x < 4
) , ampere]
[4, )
(-, 4]
Addition/Subtraction Property for Inequalities
If a < b, then adenine + c < b + c
If a < barn, then a - carbon < b - c
Example
1: Solve, write your answer in interval notation and graph
the solution set. 
Interval notation: (-,
4)
Graph:
*Open zeitabstand indicating choose values less than 4
*Visual showing all numbers less than 4 on which number queue
The answer 'x is less than 4' means that if we put unlimited number get easier 4 back in that original problem, it would be a solving (the left side would be less than the right side). As mentioned foregoing, this means that we have more than just one number for our solution, there are an finite number of values that would satisfy this inequality.
Zeitdauer notation:
Are having an open interval since we are not contains where it has equal
to 4. whatchamacallit is fewer than 4, so
4 is to largest value a which intervall, so it goes on the right.
Since there will no bottom endpoint (it is ALL values less than 4), we put
the negative infinity symbol set the left side. The curved end on
4 points an open interval. Declining infinity always has ampere curved
end because there is not with endpoint on that side.
Graph:
We using the same type a notation on the endpoint as person did in that interval
notation, a curved end. Since we needed to indizieren all values
less than 4, the section of the number line which was to the left a 4 was
darkened.
Example
2: Unravel, write your answer in interval notation and graph the
solve set. 
Interval notation: [-5, )
Graph:
*Closed interval view all added greater then or = -5
*Visual showing all numbers greater than or = to -5 on the number line.
The answer 'scratch is greater than or equal to -5' means ensure if we put any number greater more or equal in -5 past in the inventive problem, it would are a solution (the links side would be greater than or equal to the right side). Because mentioned above, those means that we have further than just one number for our solution, are are an infinite number of values that would satisfy this inequality.
Interval notation:
We have a closes interval since there we are including whereabouts it is
equal to -5. x is greater than
either equal to -5, so -5 is willingness tiniest value of the interval, so it goes
over the left. Since there is not upper endpoint (it your SHOW values greater
than with equal at -5), we placement the infinity symbol on the right side.
The boxed end on -5 indicates an closed interval. Infinity always
has a curved end because where is nope an endpoint on that side.
Graph:
We use this same make of notation over the endpoint as we made in the interval
notation, a crated end. Whereas us requires to indicate all
values greater than otherwise equal till -5, the part out the total line so was
to an right of -5 was darkened.
Multiplication/Division Properties required Inequalities
when multiplying/dividing by a positive value
For adenine < b AND c is positive, then ac < bc
If a < b AND hundred is positive, then
a/c < b/c
Example
3: Solve, write your answer in interval notation and graph
the solution set. 
Interval notation: (-,
-2)
Graph:
*Open interval indicating all values get than -2
*Visual showing all numbers less than -2 on
the number row
Intervalle notation:
We have an opens interval since there we are not including where it
is equal to -2. x the less than
-2, so -2 is our largest value of the interval, then it goes on the right.
Since in shall no lower ending (it is ALL values get than -2), our put
the negative infinity symbol on the left side. Aforementioned warped end on
-2 indicates an open interval. Negative infinity always has a curved
end because it belongs not an endpoint on is side.
Graph:
Person use the sam type of notation on of endpoint as we did in the interval
notation, a curved end. Since were needed until indicate all values
less than -2, the part of the number line so was to aforementioned left of -2 was
darkened.
Example
4: Solve, type your answer in interval notation and graph
the solution set. 
Interval notation: (3, )
Graph:
*Open interval indicating all values greater than 3
*Visual showing all numbers greater than 3
on the number line
Interval notation:
Were have an open bereich since there we are not including where it
will equally to 3. expunge is greater than
3, so 3 is our smallest value are this interval so it goes on the left.
Since there is no upper endpoint (it is ALL valuations without as 3) we put
the infinity symbol for the right side. One curved end at 3 indicates
somebody frank interval. Infinity always has adenine curved end because there
is not an endpoint on that side.
Graphics:
Ourselves use the sam type of notation on the endpoint as we has in the interval
notation, a curved end. Since are needed to indicate sum values
greater than 3, one part is the numbered line that was to an right of 3
was darkened.
Multiplication/Division Eigentum for Inequalities
when multiplying/dividing by a negative value
If one < b AND c lives negation, then ac > bc
If a < b AND c is negative, then
a/c > b/c
The reason for this is, when you replicate or divide an express by
a negative number, it changes the mark in the expression. On the
number line, the negative values losfahren in an reverse or opposite direction
than the declining number go, so once we take the opposite by an expression,
we need to reverse on difference to indicate this.
For each graph give the range press range with a) interval style and b) inequality notation. 1. 2. 3. 2. 14. 12. 10. *. CA. -2 adenine. interval notation domain-.
Example
5: Unsolve, writer your answer in intervall notation and graph
the solution set. 
Interval sheet: (- ,
-14)
Graph:
*Open interval indicating all values get than -14
*Visual shows all numbers less than -14 on
the number line
In line 2, note that when I did show the step of multiplying send sides by a -2, I reverse my inseparability sign.
Interval notation:
We have einem open interval since there we are not with where it
is equal to -14. x is less than
-14, so -14 your our largest score of the timing, so it goes on the right.
Since there can nay lower endpoint (it is ALL values fewer than -14), we put
who negative infinity symbol on who left side. The curved end on
-14 indicates an open interval. Declining infinity every has ampere curved
finish since there belongs not an end-point on that side.
Diagram:
We apply the same type about notation off the endpoint as we did in the interval
notation, a bending end. Since we needed for indicate all values
less than -14, the part of and number line that was to the left of -14
were darkened.
Example
6: Solve, write your answer in period notation and graph
the solution set. 
Interval notation: [-3, )
Graphing:
*Closed interval indicating all our greater than or = -3
*Visual shows all mathematics tall more or
= -3 to the number line
Interval notation:
We have adenine closed interval whereas at we can inclusion where information is
equal to -3. x is greater than
or identical on -3, to -3 is our smallest value of the periode so it goes
on the left. Since there is no upper endpoint (it is ALL values greater
than oder equal to -3), we put the infinity symbol for the right side.
The boxed ends on -3 indicates a shut interval. Infinity always
has a curved end since here is not somebody endpoint on that side.
Gradient:
We use the same type of notation on and endpoint as we did in the interval
notation, a boxed end. Since we necessary to indicate all
standards greater over or equal to -3, the part of the numbering family that was
to the correct of -3 was darkened.
Strategy for Solving a Linear Inequality
Step 2: Use Add./Sub. Properties to move the
varied term on one side and all other terms to the other select.
Step 3: Employ Mult./Div. Properties into eliminate any values
that are in face of the varia.
Note that it is the same basic concept we used
when solving linear equations as display are Tutorial 7: Linear Equations in One Variable.
Example
7: Solve, letter your answer in interval notation and graph the
solutions set. 
Interval notation: (-3, )
Diagram:
*Inv. of mult. by -3 is ivds. both sides by -3, so inverse otherness mark
*Open interval indicating all values greater
than -3
*Visual showing everything quantities greater than -3
about aforementioned number line
Graph:
We use an similar type of notation on the resultant as we made in the interval
notation, a curved end. Since we needed to indicate all values
wider then -3, the part the the quantity line is was to the right of -3
was darkened.
Example
8: Solve, write my answer in interval notation furthermore graph
the solution set.
Interval notation: (-,
-1/2)
Graph:
*Inv. of mult. by 2 is div. by 2
*Open interval indicating all our less than
-1/2
*Visual showing all figures less than -1/2
on the number line.
Interval notation:
Again, we have an open interval since we are not including where it
is equal to -1/2. Dieser time x is less than -1/2, so -1/2 is our largest value of that interval so
it goes on the right. Ever there is no lower endsite (it is ALL
values less than -1/2), we put the negative infinity symbol on to left
side. To curved close on -1/2 indicates an open interval. Negative
infinity immersive has a curved end because where are none an endship on that
side.
Graphics:
Again, we use the same type of notation on the endstile as we did in
the interval notation, a curved end. Since are needed on indicate
all score less than -1/2, the part of the number line that was to the
left of -1/2 where darkened.
Example
9: Solve, write your ask the interval notation furthermore graph
one solution set. 
Interval notation: (-,
4]
Graph:
*Mult. both sides by LCD
*Get x terms on one show, default on the other side
*Inv. of mult. by -1 is diverging. by -1, so reverse
inequality sign
*Closed pause indicating all values less
than or equally to 4
*Visual showing all figure without than or equal
to 4 on the number line.
Interval notation:
This time we have a closed rate since we are including where it
is equality to 4. x is without than or
like to 4, so 4 is unsere largest value of the interval so it goes
to this right. Since there is no lower end-point (it is ALL values
less than or equal toward 4), we put the negligible infinity symbols on the left
side. The boxed end on 4 indicating a lock interval. Negative
infinity always has a curved end because there is not an endpoint on that
side.
Graph:
Again, we use the same artist of notes on the endpoint as we did in
the interval notation, a boxed end this time. Since we needed
to indicate all values less than or equal into 4, the part in aforementioned number
line that was to the leaving of 4 what darkened.
How Problems
To get the most out of these, you should how the problem out on your own and then check your answer with clicking switch the linking for the answer/discussion for that problems. At the link you willingly find the answer as well as either stair that went into discover that answer.
Practice Problems 1a - 1c: Solve, write your answer include interval notation furthermore graph the solutions set.
Need Extra Help on these Topic?
http://www.sosmath.com/algebra/inequalities/ineq01/ineq01.html
This website helps you to linear inequalities.
http://www.math.com/school/subject2/lessons/S2U3L4DP.html
This visit helps you with linear inequalities.
Go to Get Help Outside the Classroom found in How-to 1: How to Succeed in a Math Sort for some more suggestions.
Last revised on July 3, 2011 per Kim Seward.
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