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Chabot Mathematics
§4.1 Solve
InEqualities
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Chabot College Mathematics
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Review § 2.5
MTH 55
 Any QUESTIONS About
• §2.5 → Point-Slope Line Equation
 Any QUESTIONS About HomeWork
• §2.5 → HW-7
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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.
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Linear InEqualities
 A linear inequality in one variable is an
inequality that is equivalent to one of the
forms that are similar to mx + b
ax  b  0 or ax  b  0
ax  b  0 or ax  b  0
 where a and b represent real numbers
and a ≠ 0.
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Example  Chk InEqual Soln

Determine whether 5 is a solution to
a) 3x + 2 >7 b) 7x − 31 ≠ 4

SOLUTION
a) Substitute 5 for x 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
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“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
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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
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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
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-4
-3
-2
-1
0
1
2
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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
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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
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Interval vs Dot Notation
 Graph x ≥ 5
Dot Graph
Interval Graph
[
 Graph x < 2
Dot Graph
Interval Graph
)
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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.
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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…
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Compact Interval Notation
 Graphed Interval Notation can be
written in Compact, 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, )
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Example  SetBuilder & Interval
 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
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]
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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
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(
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Intervals on the Real No. Line
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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;
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Example  Addition Principle
 Solve & Graph x  6  2
 Solve (get x by itself)
x66  26
x  4
Addition Principle
Simplify to Show Solution
 Graph
(
• Any number greater than −4 makes
the statement true.
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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 ≥
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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
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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}
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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
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]
30 30
Example  Add & Mult Principles
 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
]
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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}
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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}
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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
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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
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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
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Bruce Mayer, PE
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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.
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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
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)
5
x < 5, or (–∞, 5)
Bruce Mayer, PE
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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.
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Bruce Mayer, PE
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THE NON-NEGATIVE 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
always nonnegative
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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.
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34
Bruce Mayer, PE
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Example  Solve InEquality
 Solve 8x + 13 > 3x − 12
 SOLUTION
8x − 3x + 13 > 3x − 3x − 12
5x + 13 > 0 – 12
5x + 13 –13 > –12 – 13
Subtract 3x from
both sides.
Subtract 13 from
both sides.
5x > −25
5 x  25

5
5
Divide both sides
by 5 to isolate x.
x > −5
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Bruce Mayer, PE
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Example  Solve InEquality
 Solve 8x + 13 > 3x – 12
 SOLUTION Graph for x > −5
(
 SOLUTION SetBuilder Notation
{x|x > −5}
 SOLUTION Interval Notation
(−5, )
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36
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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-Prop 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
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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
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Example  AirCraft E.T.A.
 Translate the InEquality for Worry
Plane’s distance
from Miami
≥
Distance from
Miami to
Bermuda
150  300t  1035
150  300t  150  1035  150
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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 about 3 hours
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40
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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?
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Bruce Mayer, PE
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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
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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
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43
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Example  CellPhone $Budget
 Carry
Out
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44
Bruce Mayer, PE
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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.
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WhiteBoard Work
 Problems From §4.1 Exercise Set
• 62 (ppt), 53, 72, 80

Working Thru
a Linear
InEquality
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P4.1-62
 Write
InEquality for
Passion
greater-than,
or equal-to
Intimacy
 Find Crossing
Point
 Thus Ans
[0,5) or
{ x x  5}
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All Done for Today
Eric
Heiden
Won Five Gold Medals and Set Five Olympic Records at the
1980 Winter Olympics
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Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
–
Chabot College Mathematics
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