1.06 Linear Inequalities and Properties of Inequalities from Intermediate Algebra: One Step at a Time © 1998 Pages 77-90 Dr. Robert J. Rapalje Seminole State College of Florida THIS SECTION IS UNDER CONSTRUCTION!
(Send me an Email if you need
this one!!) In
describing equations and inequalities, the Trichotomy
Axiom and interval notation will be helpful.
Consider the variable x and any number, for example,
4. As the word
"tri" means "three," according to the Trichotomy Axiom,
there are three ways to compare the variable x to the
number 4: x=4, x<4, or
x>4.
x<4
x=4 x>4 The
first category x=4 consists
of just one point. The second
category x<4
represents an entire interval of numbers on the number line to the left of
the number 4. Likewise, the
third category x>4
represents an entire interval of numbers on the number line to the
right of the number 4. Notice
in each of these cases, the endpoint 4 is not included.
If the endpoint 4 is to be included, then you write x
≤ 4 or x
≥ 4 respectively. Remember
that x < 4 is
equivalent to 4 > x,
and that x
≤ 4 is equivalent to 4
≥ x. Suppose
the variable x represents all values between -3 and 4.
In this case, x must be greater than -3 and at the
same time less than 4. Another
way to write this is -3 is less than x and at the same time x
is less than 4. In math
symbols, this is -3<x
and x<4. This
can be written -3<x<4,
which in English just means that “x is between -3 and 4, not including the endpoints.” Since
in each of these intervals
there are infinitely many numbers, it is helpful to describe the interval
with interval notation (illustrated on the next page!).
Interval notation is always given from left
to right, with brackets "["
or "]" indicating included
endpoints, and parentheses "("
or ")" indicating
that endpoints are not included.
If the interval extends all the way to the right
(infinity) or to the left (negative infinity), the symbols “∞ ”
or “- ∞ ” are used. The
tradition has always been to consider - ∞
and ∞ non-inclusive, since infinity is not something that can be
"contained" or "included." Variable
Notation
Interval Notation
Graph on Numberline
0
1 2
3 4
0 1 2
3 4
0
1 2 3
4
0
1 2 3
4
-3
-2 -1
0 1 2
3 4
-3 -2 -1 0
1 2
3 4
-3
-2 -1 0
1 2
3 4
-3
-2 -1
0 1
2 3 4 Realize
that inequalities, like equations, must often be solved.
Remember that when you solved equations, according to the addition
rule for equations, you were allowed to add (or subtract) the same
number from both sides of an equation.
Also, according to the multiplication
rule for equations, you were allowed to multiply (or divide) both
sides of an equation by the same number.
Wouldn’t it be nice if the same rules applied to inequalities as
well as equations? Well, it is almost that simple, but not quite.
It will be necessary to modify the multiplication rule slightly. Take
the inequality
-2 < 4, and add +2 to each side
+2
+2
0 < 6
Still true! Use
the same inequality
-2 < 4, and add -2 to each side
-2 -2
-4 < 2
Still true! Use
the same inequality and multiply (or divide) both sides by +2.
-2 <
4
+2(-2)<
+2(4)
- 4 < 8 Still
true! Now
try multiplying (or dividing) both sides by a negative, say -2.
-2 <
4
-2(-2) ?? -2(4)
+4
> -8 INEQUALITY MUST BE REVERSED!
These
examples verify (but do not
prove!) the following rules:
In
summary, remember that these are the same rules as the ones used to solve
equations, except that when
you multiply or divide both sides of an inequality by a negative number,
you must change the direction of the inequality sign.
Remember also that x>4 means the same as 4<x.
The expression -2<x<6 means that x
represents any number between -2 and 6, and the expression
6>x>-2 means the same as -2<x<6.
However, the notation -2<x<6, from smallest to
largest (left to right), is preferred. In
the following examples,
solve the inequalities. Give
answers in interval notation:
EXAMPLE 1
EXAMPLE 2
7x + 12 ≥ 5x
+ 4
5x - 12 >
7x + 4
-5x -5x
-7x -7x
2x
+ 12 ≥
+ 4
-2x - 12
> + 4
-
12
-12
+
12 +12
2x
≥
-8
-2x
> 16
x ≥ -4 (DO NOT REVERSE!)
x <
-8 (REVERSE
SIGN!)
[-4, ∞ )
(-∞, -8)
EXERCISES: Solve for x.
Give answers in interval notation. 1.
5 + 3x
≤ x
- 3
2. – 3x + 4 > 4x
- 3 3.
5 – 3x <
x - 3
4. – 3x + 15 ≥ 3x
- 3 Also,
you should be aware that inequalties, like equations, can be conditional,
identities, or contradictions.
EXAMPLE 3
EXAMPLE 4
5x + 12 ≥ 5x
+ 20
5x - 12 < 5x
+ 20
-5x
-5x
-5x
-5x
12 ≥
+ 20
- 12 < +
20
Never true
True
for all x
NO
SOLUTION
(-∞ , ∞
)
or
EMPTY SET EXERCISES: Solve for x.
Give answers in interval notation. 5.
5 - 3(x - 4) ≤ 2(x
- 4)
6. - 3(x
+ 4) > 4(2x + 3)
– 5x 7.
2x - 8(7 - x)
> 5(2x - 4)
8. -8
- 3(x - 4) < 3(6 - x) 9.
2x - 8 <
2(2 – 5x) + 12x
10.
-2(2 + 3x) ≥
3(5 - x) + 8
EXAMPLE 5
EXAMPLE 6
- 5 ≤ 2x
+ 3 ≤
7
- 1 < 3 – 2x
≤
5
- 3
- 3 -3
-
3 -3 -3
- 8 ≤
2x ≤
4
- 4 < –2x
≤
2
- 4 ≤
x ≤
2
2 >
x ≥
-1
[-4,
2]
[-1, 2)
EXAMPLE 7
EXAMPLE 8
‑
15 ≤ 2x+3 ≤
21
- 3 < 3 – 2x
≤ 15
- 18
≤ 2x
≤ 18
- 6 < -2x
≤ 12
- 9 ≤
x ≤ 9
3 > x
≥ -6
[-9, 9]
[-6, 3) 11.
–7 ≤
3 ‑ 2x < 5
12. –
5 <
2x +
3 < ‑1 13.
15.
FORMAL EXPLANATION As
an introduction to a more formal explanation of
inequalities, consider again the properties of equations from section
1.04. Which, if any, of these
properties of equations apply also to inequalities?
ADDITION PROPERTY FOR EQUATIONS
If a=b, then a
+ c = b + c
If a=b, then a - c = b - c.
The same number
may be added (or subtracted) from both sides
of an equation. I. ADDITION
PROPERTY FOR INEQUALITIES
If a < b, then a + c < b + c
If a < b,
then a - c < b - c
If
a > b, then a + c > b + c
If a > b,
then a - c > b - c. The
addition property for inequalities is basically the same as it is for
equations--the same number may be added (or subtracted) from both sides of
an inequality, and the inequality remains the same.
Of course, this property is also valid for "≤
" and "≥ ."
MULTIPLICATION PROPERTY FOR EQUATIONS
If a=b, then ac
= bc
If a=b and c ≠ 0, then a/c = b/c.
Both sides of an equation may be multiplied or divided by the same
non-zero number. II.
MULTIPLICATION
PROPERTY FOR INEQUALITIES
must
be given in two distinct parts:
A.
If a < b and c > 0, then
ac < bc.
If a > b and c > 0, then
ac > bc.
If a ≤
b and c > 0, then ac
≤ bc.
If a ≥
b and c > 0, then ac
≥ bc. This
means that if both sides of an inequality are multiplied (or divided) by a
positive number, then the inequality sign remains the same.
B. If
a < b and c < 0, then ac
> bc.
If a > b and c < 0, then
ac < bc.
If a ≤ b
and c < 0, then ac ≥
bc.
If a ≥
b and c < 0, then ac
≤ bc. This
means that if both sides of an inequality are multiplied (or divided) by a
negative number, then the inequality sign must be reversed. REFLEXIVE
PROPERTY FOR EQUATIONS: a
= a. Any number is equal to itself.
Clearly, a number is not
less than or greater than itself. Therefore,
there is NO reflexive
property for inequalities. SYMMETRIC PROPERTY FOR
EQUATIONS: If a = b, then b =
a. The order in which
the equality is given does not matter.
For example, you can say "x=4"
or "4=x", the meaning is the same--the value of x
is 4. Clearly,
if a < b, then it is not true that b < a. There is NO
symmetric property for inequalities. TRANSITIVE PROPERTY FOR
EQUATIONS: If a = b and b =
c, then a = c. The word "trans" means "across."
If you can get from point "a" to "b", and then
from "b" to "c", then you can get from "a" across
"b" to "c." III.
TRANSITIVE PROPERTY FOR INEQUALITIES.
If a < b and
b < c, then a < c.
If a > b and b > c, then a > c.
The transitive property is also valid for "≤
" and "≥ ." EXERCISES:
In each of the following exercises,
fill in the blank with the name of the property. 1.
If x + 3 = 5, then x
= 2 ___________________________________
because
you added -3 to both sides of the equation.
2.
If x - 3 = 5, then x
= 8 ___________________________________
because you
added 3 to both sides of the equation.
3.
If x + 3 < 5, then x
< 2 ___________________________________
because you
added -3 to both sides of the inequality.
4.
If x + 3 > 5, then x
> 2 _________________________________
__
because
you added -3 to both sides of the inequality.
5.
If 7x
= 21, then x = 3 ____________________________
_______
because you
multiplied both sides of the equation by 1/7.
6.
If 7x
> 21, then x >
3 ________________________________
___
because
you multiplied both sides of the inequality by 1/7.
7.
If 7x < 21, then x
< 3 ________________________________
__
because
you multiplied both sides of the inequalitty by 1/7.
8.
If -7x > 21, then x
<-3
because you
multiplied both sides of the inequality by -1/7.
9.
If -7x = 21, then x
=-3 ______________________________
____
because you multiplied both sides of
the equation by -1/7. 10.
If -7x
< 21, then x
>-3 _____________________________
_____
because
you multiplied both sides of the inequality by -1/7. 11.
If x = 3 and 3 = y, then x = y
_____ _
____ property of ______
____ 12.
If 5 > 3 and 3 > 1, then 5 > 1
_____ _
____ property of ______
____
13.
If -3< 3 and 3 < 6, then -3< 6
_____
_ ____
property of ______ ____
14.
If x < 3 and 3 < y, then x < y
_____ _
____ property of ______
____ 15.
If 4x > 20, then x
> 5
__________________
_________ 16.
If 4x > 20 and 20 > 16, then 4x>16 ______
_____________________ 17.
If x + 4 < 12, then x
< 8
_________________
__________ 18.
If x + 4 = 12, then x
= 8
__________________________
_ 19.
If 4x = 20, then x
= 5
_________________
_________ 20.
If 4x = 20 and 20 = 2y, then 4x=2y _________________________
__ 21.
If –4x> 20, then x
< -5
___________________
________ 22.
If x
> -5, then 4x >
-20
___________________
________ 23.
If x
+ 4 = 12, then 2x +
8 = 24 _______________
____________ 24.
If x
= 12, then x + 8 =
20
________________
___________ 25.
If x
= 12, then 12 = x _________________
__________ 26.
6 = 6
___________________
________ 27.
If 12 < x and x
< 20, then 12<20 _______________________
____ 28.
If -x < 12, then x
> -12
________________________
___ 29.
If 3x + 12 > 27, then 3x > 15
____________________
_______ 30.
If 3x > 15, then x
> 5
_________________________
__ COMPOUND INEQUALITIES Before
introducing the idea of compound inequalities (two inequalities connected
with the by the words “and” or
“or”) it will be helpful to make use of set notation, and define the words “and” and “or”.
A I B A
“intersection” B) is the set containing all members that are common to
both set A and set B. This
consists of the overlap of the sets A and B.
It is like the “intersection” of two streets or highways--where
the two streets come together and overlap.
A U B (A
“union” B) is the set containing all members that are in either set A or
set B or both. The union consists of everything included in either set.
In terms of streets or highways, the union includes both entire
streets. Suppose
you were given two sets of numbers as follows:
A =
{2,4,6,8}
and
B = {1,4,6,8}
A
I B = {4,6,8} (Include
only the overlap of A and B!)
A
U B = {1,2,4,6,8} (Include everything that is in A or B or in both!) Frequently
two simple statements (equations or inequalities) are connected with the
words “and” or “or” to form a compound
statement. When the word “and”
is used, the statement is called a conjunction,
and the solution is the intersection
of the two solution sets. When
the word “or” is used, the
statement is called a disjunction,
and the solution is the union
of the two solution sets.
In
each of the following examples and exercises,
you will be asked to solve compound inequalities with the key words “and” or “or”.
The answers of course will be graphed on a numberline and given
in interval notation.
The secret to success in this (learned after more than 20 or 25
years of experience!) is this: Draw a numberline, but then
do the problem ABOVE THE NUMBERLINE (NOT ON THE NUMBERLINE--SAVE
THE NUMBERLINE FOR YOUR FINAL ANSWER!
The following examples illustrate this.
Remember, do the problem in the space provided above the
numberline, then bring the intersection
or union down on the numberline, and finally give the answer in interval
notation.
EXAMPLE 1
EXAMPLE 2 x
> 5 and
x ≥ – 2
x > 5 or
x ≥
– 2
.
. . -3 -2 -1 0 1 2 3 4 5 6 . . .
. . . –3
-2 -1
0 1 2
3 4
5 6 . . .
ANSWER:
(5, ∞ )
ANSWER:
[–2, ∞ )
EXAMPLE
3
EXAMPLE 4
x < 5 and
x ≥
– 2 x < 5 or
x ≥
– 2
[
[
. . . -3 -2
–1 0 1 2 3 4 5 6 . . .
. . . -3 -2 -1 0 1 2 3 4 5 6 . . .
ANSWER: [– 2, 5)
ANSWER:
(– ∞ , ∞ )
This is called “BETWEENNESS.”
This may also be written
“All Real
Numbers.”
EXAMPLE 5
EXAMPLE 6
x > 5 and
x ≤
– 2
x
> 5 or x
≤ – 2
. . . -3 -2 -1
0 1 2 3 4 5 6 . . .
. . . -3 -2 -1 0 1 2 3 4 5 6 . . .
ANSWER:
No Solution
ANSWER: Two separate intervals
Empty Set { }
(- ∞, – 2] U (5, ∞ )
This is called “EXTREMES.” EXERCISES:
Solve the compound inequalities and give interval notation. 1. x
< 4 and x
≤ – 2
2.
x
< 4 or x
≤ – 2
________________________________
_________________________________
.
. .
–3 –2 –1
0 1
2 3 4
5 6
. . .
. . .
–3 –2
–1 0 1
2 3
4 5 6
. . .
ANSWER:
ANSWER: 3. x
> 4 and x
≤ – 2
4.
x
> 4 or
x ≤
– 2
________________________________
_________________________________
.
. .
–3 –2 –1
0 1
2 3 4
5 6
. . .
. . .
–3 –2
–1 0 1
2 3
4 5 6
. .
.
ANSWER:
ANSWER: 5.
x
< 4
or x ≥ –
2 6.
x
< 4
and x ≥
– 2
________________________________
_________________________________
.
. .
–3 –2 –1
0 1
2 3 4
5 6
. . .
. . .
–3 –2
–1 0 1
2 3
4 5 6
. .
.
ANSWER:
ANSWER: 7.
x
≥ 4
or x ≥ –
2
8.
x
≥ 4
and x
≥ – 2
________________________________
_________________________________
.
. .
–3 –2 –1
0 1
2 3 4
5 6
. . .
. . .
–3 –2
–1 0 1
2 3
4 5 6
. .
.
ANSWER:
ANSWER: 9.
x < – 4
or x ≥ – 2
10.
x
< –4 and x
≥ – 2
________________________________
_________________________________
.
. .
–3 –2 –1
0 1
2 3 4
5 6
. . .
. . .
–3 –2
–1 0 1
2 3
4 5 6
. .
.
ANSWER:
ANSWER: 11. x
> – 5 and x
≤ – 2
12. x
> – 5
or x
≤ – 2
________________________________
_________________________________
.
. .
–3 –2 –1
0 1
2 3 4
5 6
. . .
. . .
–3 –2
–1 0 1
2 3
4 5 6
. .
.
ANSWER:
ANSWER: 13. – 4x
< 4 or x
– 7 ≤ – 2
14. – 4x
< 4 and x
– 7 ≤ – 2
________________________________
_________________________________
.
. .
–3 –2 –1
0 1
2 3 4
5 6
. . .
. . .
–3 –2
–1 0 1
2 3
4 5 6
. .
.
ANSWER:
ANSWER: 15. – 4x
≥
4 or
– x ≥ – 2
16. –
4x
≥ 4 and
– x ≤ – 2
________________________________
_________________________________
.
. .
–3 –2 –1
0 1
2 3 4
5 6
. . .
. . .
–3 –2
–1 0 1
2 3
4 5 6
. .
.
ANSWER:
ANSWER: 17.
–
2x – 6 < – 4 or
3x – 6
≥ – 6
18.
– 2x < – 4 and
3x – 6
≥ – 6 19. x–3
> – 5 and – 3x
≥ – 6
20.
x – 3 <
– 5 or
– 3x – 6 ≤ –
6 ANSWERS
1.06 p.
80-82:
1. (-∞,
-4]; 2. (-∞
, 1);
3. (2,
∞ ); 4. (-∞ , 3]; 5. [5, ∞
); 6. (-∞,
-4); 7.
No Solution or { };
8.
(-∞
, ∞
); 9. (-∞
,
∞);
10.(-
∞,-9]; 11. (-1,
5]; 12.
(-4,-2); 13. (-6, 12]; 14.
(-9,-3); 15.
[-2,4];
16. [-3,7). p.
84-85:
1. Add prop
for eq; 2. Add prop for eq; 3. Add
prop for ineq; 4. Add prop for ineq;
5.
Mult prop for eq; 6. Mult
prop for ineq; 7. Mult ineq by
positive; 8. Mult ineq by
negative;
9. Mult prop for eq; 10. Mult ineq by negative; 11.
Transitive, equations; 12. Transitive,
inequalities; 13.
Transitive, inequalities; 14. Transitive, inequalities; 15.
Mult
ineq by positive; 16.
Transitive, inequalities; 17. Addition prop for ineq; 18. Addition
prop for eq; 19. Mult prop for
eq; 20. Transitive, equations;
21. Mult ineq by neg; 22. Mult
ineq by positive; 23. Mult prop
for eq; 24.
Add prop for eq; 25. Symmetric for eq; 26.
Reflexive
for eq; 27.
Transitive for ineq; 28.
Mult ineq by negative;
29.
Add
prop for ineq; 30. Mult ineq by positive.
p.
88-90:
1. (-∞
, -2]; 2. (-∞,4);
3. No
Solution; 4. (-∞, -2] U (4, ∞); 5.
(-∞, ∞);
6. [-2,4); 7. [-2,
∞); Return to top of page Return to main page Seminole State main page
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