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Chabot Mathematics
§4.1 Solve
InEqualities
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Chabot College Mathematics
1
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Review § 3.3
MTH 55
 Any QUESTIONS About
• §3.3b → 3 Variable Linear System
Applications
 Any QUESTIONS About HomeWork
• §3.3b → HW-09
Chabot College Mathematics
2
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Solving InEqualities
 An inequality is any sentence containing
, , ,  , or  .
 Some
3x  2  7, c  7, and 4x  6  3.
Examples
 Any value for a variable that makes an
inequality true is called a solution. The
set of all solutions is called the solution
set. When all solutions of an inequality
are found, we say that we have solved
the inequality.
Chabot College Mathematics
3
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Linear InEqualities
 A linear inequality in one variable is an
inequality that is equivalent to one of the
forms
ax  b  0 or ax  b  0
 where a and b represent real numbers
and a ≠ 0.
Chabot College Mathematics
4
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Chk InEqual Soln

Determine whether 5 is a solution to
a) 3x + 2 >7 b) 7x − 31 ≠ 4

SOLUTION
a) Substitute to get 3(5) + 2 > 7, or 17 >7, a
true statement. Thus, 5 is a solution to
InEquality-a
b) Substitute to get 7(5) − 31 ≠ 4, or 4≠ 4, a
false statement. Thus, 5 is not a
solution to InEquality-b
Chabot College Mathematics
5
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
“Dot” Graphs of InEqualities
 Because solutions of inequalities like
x < 4 are too numerous to list, it is
helpful to make a drawing that
represents all the solutions
 The graph of an inequality is such a
drawing. Graphs of inequalities in one
variable can be drawn on a number line
by shading all the points that are
solutions. Open dots are used to
indicate endpoints that are not solutions
and Closed dots are used to indicated
endpoints that are solutions
Chabot College Mathematics
6
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Graph Inequalities
 Graph Inequalities:
a) x < 3, b) y ≥ –4; c) –3< x ≤ 5
 Soln-a) The solutions of x < 3 are those
numbers less than 3.
• Shade all points to the left of 3
-4
-3
-2
-1
0
1
2
3
4
5
6
• The open dot at 3 and the shading to the
left indicate that 3 is NOT part of the graph,
but numbers such as 1 and –2 are
Chabot College Mathematics
7
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Graph Inequalities
 Graph Inequalities:
a) x < 3, b) y ≥ –4; c) –3< x ≤ 5
 Soln-b) The solutions of y ≥ –4 are
shown on the number line by shading
the point for –4 and all points to the
right of –4.
• The closed dot at –4 indicates that –4 IS
part of the graph
-7
-6
-5
Chabot College Mathematics
8
-4
-3
-2
-1
0
1
2
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
3
Example  Graph Inequalities
 Graph Inequalities:
a) x < 3, b) y ≥ –4;
c) −3< x ≤ 5
 Soln-c) The inequality −3 < x  5 is read
“−3 is less than x, AND x is less than
or equal to 5.”
-5
-4
-3
-2
-1
0
1
2
3
4
• Note the
– OPEN dot at −3 → due to −3< x
– CLOSED dot at 5 → due to x≤5
Chabot College Mathematics
9
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
5
Interval Notation
 Interval Notation for Inequalities on
Number lines can used in Place of
“Dot Notation:
• Open Dot, ‫ → ס‬Left or Right, Single
Parenthesis
• Closed Dot, ● → Left or Right, Single
Square-Bracket
Chabot College Mathematics
10
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Interval vs Dot Notation
 Graph x  5
Dot Graph
Interval Graph
[
 Graph x < 2
Dot Graph
Interval Graph
)
Chabot College Mathematics
11
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Interval Graphing of InEqualities
 If the symbol is  or , draw a bracket
on the number line at the indicated
number. If the symbol is < or >, draw a
parenthesis on the number line at the
indicated number.
 If the variable is greater than the
indicated number, shade to the right of
the indicated number. If the variable is
less than the indicated number, shade
to the left of the indicated number.
Chabot College Mathematics
12
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Set Builder Notation
 In MTH55 the INTERVAL form is
preferred for Graphing InEqualities
 A more compact alternative to
InEquality Solution Graphing is
SET BUILDER notation:
x | x  3
SET BUILDER
Notation
Read as: “x such that x is…
Chabot College Mathematics
13
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Compact Interval Notation
 Graphing Interval Notation can be
written in ShortHand form by
transferring the Parenthesis or Bracket
from the Graph to Enclose the
InEquality.
 Examples
• x  13 → (−, 13]
• −11< x  13 → (−11, 13]
• −11< x → (−11, )
Chabot College Mathematics
14
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Set Builder
 Write the solution set in set-builder
notation and interval notation, then
graph the solution set.
a) x  −2
b) n > 3
 SOLUTION a)
• Set-builder notation: {x|x  −2}
• Interval notation: (−, −2]
• Graph
Chabot College Mathematics
15
]
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Set Builder
 Write the solution set in set-builder
notation and interval notation, then
graph the solution set.
a) x  −2
b) n > 3
 SOLUTION b)
• Set-builder notation: {n|n > 3}
• Interval notation: (3, )
• Graph
Chabot College Mathematics
16
(
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Intervals on the Real No. Line
Chabot College Mathematics
17
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Addition Principle for InEqs
 For any real numbers a, b, and c:
• a < b is equivalent to a + c < b + c;
• a ≤ b is equivalent to a + c ≤ b + c;
• a > b is equivalent to a + c > b + c;
• a ≥ b is equivalent to a + c ≥ b + c;
Chabot College Mathematics
18
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Addition Principle
 Solve & Graph x  6  2
 Solve
x66  26
x  4
Addition Principle
Simplify to Show Solution
 Graph
(
• Any number greater than −4 makes
the statement true.
Chabot College Mathematics
19
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Addition Principle
 Solve & Graph 4 x 1  x 10
 SOLUTION
4 x 1  1  x 10  1
Add ONE to Both sides
Simplify
4x  x  9
Subtract x from Both Sides
4 x  x  x  x   9
1
Divide Both Sides by 3
3x  9
3
x  3
Simplify & Show Solution
]
Chabot College Mathematics
20
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Multiplication Principle for InEqs
 For any real numbers a and b,
and for any POSITIVE number c:
• a < b is equivalent to ac < bc, and
• a > b is equivalent to ac > bc
 For any real numbers a and b,
and for any NEGATIVE number c:
• a < b is equivalent to ac > bc, and
• a > b is equivalent to ac < bc
 Similar statements hold for ≤ and ≥
Chabot College Mathematics
21
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Mult. Principle Summarized
 Multiplying both Sides of an
Inequality by a NEGATIVE
Number REVERSES the
DIRECTION of the Inequality
• Examples
 3x  6  2 3  3x 18  6
4x 1  x 101  4x  1   x  10
Chabot College Mathematics
22
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Solve & Graph
 Solve & Graph
a)  4 y  20
 Soln-a)  4 y
20

4 4
 Graph
y  5
1
b)
x4
7
Divide Both Sides by -4
Reverse Inequality as the
Eqn-Divisor is NEGATIVE
(
 The Solution Set: {y|y > −5}
Chabot College Mathematics
23
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Solve & Graph
1
 Soln-b) x  4
7
1
7 x  47
7
x  28
Multiply Both Sides by 7
Simplify
 Graph
5 5
10 10
15 15
20 20
25 25
 The Solution Set: {x|x  28}
Chabot College Mathematics
24
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
]
30 30
Example  Solve 3x − 3 > x + 7
 Soln 3x  3  3  x  7  3 Add 3 to Both Sides
3x  x  10 Simplify
3x  x  x  x  10 Subtract x from Both Sides
2 x  10
Divide Both Sides by 2
2
Simplify
x5
 Graph
-2
-1
0
1
2
3
4
(
5
6
7
 The Solution Set: {x|x > 5}
Chabot College Mathematics
25
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
8
Example  Solve 15.4 − 3.2x < − 6.76
 Soln 10015.4  3.2 x  6.76 To Clear Decimals
100 15.4 100  3.2 x  100 6.76 Dist. in the 100
1540  320x  676 Simplify
1540 1540  320x  676 1540 Subtract 1540
 1 

 320 x  2216 Simplify; Mult. By −1/320
 320 
 2216
x
 6.925 Simplify; note that Inequality
REVERSED by Neg. Mult.
 320
 The Solution Set: {x|x > 6.925}
Chabot College Mathematics
26
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Solve & Graph
 Solve 5x  3  7 x  4x  3  9
 Soln
Use Distributive Law to
5x 15  7 x  4x 12  9 Clear Parentheses
 2 x 15  4 x  3 Simplify
 2x 15  3  4x  3  3 Add 3 to Both Sides
 2x 12  4x
Simplify
2x  2 x 12  2 x  4x Add 2x to Both Sides
 1  Simplify; Divide
 12  6 x   Both Sides by 6
6
Chabot College Mathematics
27
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Solve & Graph
 Solve 5x  3  7 x  4x  3  9
1
 Soln  12  6 x    2  x From Last Slide
6
x  2
Put x on R.H.S.; Note Reversed Inequality
 Graph
-8
-7
-6
-5
-4
-3
]
-2
-1
0
1
 The Solution Set: {x|x  –2}.
Chabot College Mathematics
28
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
2
Equation ↔ Inequality
Equation
Replace
= by
Inequality
x=5
<
x<5
3x + 2 = 14
≤
3x + 2 ≤ 14
5x + 7 = 3x +
23
>
5x + 7 > 3x +
23
x2 = 0
≥
x2 ≥ 0
Chabot College Mathematics
29
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Terms of the (Inequality) Trade
 An inequality is a statement that one
algebraic expression is less than, or is less
than or equal to, another algebraic expression
 The domain of a variable in an inequality is
the set of all real numbers for which BOTH
sides of the inequality are DEFINED.
 The solutions of the inequality are the real
numbers that result in a true statement when
those numbers are substituted for the variable
in the inequality.
Chabot College Mathematics
30
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Terms of the (Inequality) Trade
 To solve an inequality means to find all
solutions of the inequality – that is, the
solution set.
• The solution sets are intervals, and we
frequently graph the solutions sets for
inequalities in one variable on a number line
• The graph of the inequality x < 5 is the
interval (−, 5) and is shown here
Chabot College Mathematics
31
)
5
x < 5, or (–∞, 5)
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Terms of the (Inequality) Trade
 A conditional inequality such as x < 5
has in its domain at least one solution
and at least one number that is not a
solution
 An inconsistent inequality is one in
which no real number satisfies it.
 An identity is an inequality that is
satisfied by every real number in the
domain.
Chabot College Mathematics
32
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
THE NONNEGATIVE IDENTITY
x 0
2
for ANY real number x
 Because x2 = x•x is the product of either
(1) two positive factors, (2) two negative
factors, or (3) two zero factors, x2 is
always either a positive number or zero.
That is, x2 is never negative, or is
nonnegative
Chabot College Mathematics
33
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Solving Linear InEqualities
1. Simplify both sides of the inequality as needed.
a. Distribute to clear parentheses.
b. Clear fractions or decimals by multiplying through
by the LCD just as was done for equations.
(This step is optional.)
c. Combine like terms.
2. Use the addition principle so that all variable terms
are on one side of the inequality and all constants
are on the other side. Then combine like terms.
3. Use the multiplication principle to clear any
remaining coefficient. If you multiply (or divide) both
sides by a negative number, then reverse
the direction of the inequality symbol.
Chabot College Mathematics
34
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Solve InEquality
 Solve 8x + 13 > 3x − 12
 SOLUTION
8x − 3x + 13 > 3x − 3x − 12
5x + 13 > 0 – 12
x + 13 –13 > –12 – 13
Subtract 3x from
both sides.
Subtract 13 from
both sides.
5x > −25
5 x  25

5
2
Divide both sides
by 5 to isolate x.
x > −25
Chabot College Mathematics
35
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  Solve InEquality
 Solve 8x + 13 > 3x – 12
 SOLUTION Graph for x > −25
(
 SOLUTION SetBuilder Notation
{x|x > −25}
 SOLUTION Interval Notation
(−25, )
Chabot College Mathematics
36
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  AirCraft E.T.A.
 An AirCraft is 150 miles along its path
from Miami to Bermuda, cruising at 300
miles per hour, when it notifies the tower
that The Twin-Turbo is now
set on automatic pilot.
 The entire trip is 1035 miles, and we
want to determine how much time we
should let pass before we become
concerned that the plane has
encountered Bermuda-Triangle trouble
Chabot College Mathematics
37
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  AirCraft E.T.A.
 Familiarize
• Recall the Speed Eqn:
Distance = [Speed]·[time]
• So LET t ≡ time elapsed since plane on
autopilot
 Translate
• 300t = distance plane flown in t hours on
AutoPilot
• 150 + 300t = plane’s distance from Miami
after t hours on AutoPilot
Chabot College Mathematics
38
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  AirCraft E.T.A.
 Translate the InEquality
Plane’s distance
from Miami
≥
Distance from
Miami to
Bermuda
150  300t  1035
150  300t  150  1035  150
Chabot College Mathematics
39
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  AirCraft E.T.A.
 Carry Out
300t  885
300t 885

300 300
t  2.95
 State: Since 2.95 is roughly 3 hours, the
tower will suspect trouble if the plane
has not arrived in 3 hours
Chabot College Mathematics
40
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  CellPhone $Budget
 You have just purchased a new cell
phone. According to the terms of your
agreement, you pay a flat fee of $6 per
month, plus 4 cents per minute for calls.
 If you want your total bill to be no more
than $10 for the month, how many
minutes can you use?
Chabot College Mathematics
41
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  CellPhone $Budget
 Familiarize: Say we use the phone 35
min per month. Then the Expense
$6
$0.04 35 min
$7.4



month
min
month month
 Now that we understand the calculation
LET
• x ≡ CellPhone usage in minutes per month
Chabot College Mathematics
42
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  CellPhone $Budget
 Translate:
Montly 

 Plus
Expense 
Minute  To Be


 
 $10
Expense  Less Than 
$6  0.04 x  $10
4
 Or, With
6
x  10
0.04 = 4/100
100
Chabot College Mathematics
43
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  CellPhone $Budget
 Carry
Out
Chabot College Mathematics
44
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Example  CellPhone $Budget
 Check: If the phone is used for 100
minutes, you will have a total bill of
$6 + $0.04(100) or $10 
 State: If you use no more than 100
minutes of cell phone time, your bill will
be less than or equal to $10.
Chabot College Mathematics
45
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
WhiteBoard Work
 Problems From §4.1 Exercise Set
• 72 (ppt), 62, 82, 90

Working Thru
a Linear
InEquality
Chabot College Mathematics
46
Bruce Mayer, PE
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P4.1-72
 Write
InEquality for
Passion
greater than,
or equal to
Intimacy
 Find Crossing
Point
 Thus Ans
[0,5) or
{ x x  5}
Chabot College Mathematics
47
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
All Done for Today
Eric
Heiden
Won Five Gold Medals and Set Five Olympic Records at the
1980 Winter Olympics
Chabot College Mathematics
48
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
–
Chabot College Mathematics
49
Bruce Mayer, PE
[email protected] • MTH55_Lec-14_sec_3-3a_3Var_Sys_Apps.ppt