Please close your laptops
and turn off and put away your
cell phones, and get out your
note-taking materials.
Today’s daily 5-minute quiz will
be given at the end of class.
Weekly Quiz 2 Results:
• Average class score after partial credit: _______
(_____ raw score)
• Commonly missed questions: #_______________
Grade Scale
Grade
A
A-
B+
B
B-
C+
C
C-
F
Points
≥ 920 ≥ 890 ≥ 860
≥ 830
≥ 800
≥ 750
≥ 700 ≥ 670 < 670
% Score
≥ 92
≥ 83
≥ 80
≥ 75
≥ 70
≥ 89
≥ 86
≥ 67
< 67
Why you should keep taking the practice
quiz until you can score at least 90%:
(B+)
(almost a C-)
(low F)
(A)
REMINDERS for the upcoming Test 1:
• Take the practice test early enough so you’ll
have time to review it, retake it, come into
the open lab for help if needed.
• Review each practice test after you submit it.
(The “help me solve this” buttons will appear when you review the test.)
• You have unlimited attempts, so retake the
practice test until you score at least 90%.
• If you score < 90%, come into the open lab
to review your practice test with a TA.
(Or just take the practice test in the open lab to start with …)
Note to teachers: You can use item analysis
to see which questions your section missed
most, and you can insert slides here with
screen shots of those questions you want
to go over in class.
Quiz question # ____ (xx%)
Section 2.8
Linear Inequalities 1
Linear Inequalities
An inequality is a statement that contains one of
the symbols: < , >, ≤ or ≥.
Linear equations:
Linear inequalities:
x=3
x>3
12 = 7 – 3y
12 ≤ 7 – 3y
Graphing solutions to linear inequalities
in one variable
• Use a number line.
• Use a square bracket at the endpoint of an interval if you
want to include the point.
• Use a parenthesis at the endpoint if you DO NOT want to
include the point.
Graph the inequality x  7:
Graph the inequality x > – 4:
Using graphs to figure out how to write
a solution in interval notation:
]
-∞
7
-∞
∞ The inequality x  7
is expressed in interval
notation as (-, 7]
(
∞ The inequality x > -4 is
-4
expressed in interval
notation as (-4, )
IMPORTANT:
In interval notation, ∞ and -∞
ALWAYS are enclosed by
a (round bracket)
NEVER by a [ square bracket].
Example from today’s homework:
x  9}
[9, )
Addition property of inequality
• a< b and a + c < b + c are equivalent inequalities.
Example: 2 ≤ 4
and 2 + (-3) ≤ 4 + (-3)
are equivalent
Multiplication property of inequality
• if c is positive, then:
a< b and ac < bc are equivalent inequalities,
Example: 3 ≥ 1 (multiply both sides by 2); so 6 ≥ 2 is equivalent.
• if c is negative, then:
a< b and ac > bc are equivalent inequalities,
Example: 3 ≥ 1 (multiply both sides by -2); so -6 ≤ -2 is equivalent.
.
Solving linear inequalities in one variable
1)
2)
3)
4)
Multiply to clear fractions.
Use the distributive property (parentheses).
Simplify each side of the inequality.
Get all variable terms on one side and
numbers on the other side of inequality
(addition property of inequality).
5) Isolate variable by dividing both sides by the
number in front of the variable (multiplication
property of inequality).
6) Do not forget to change the direction of the
inequality sign if you multiply or divide
both sides by a negative number.
Don’t forget that if both sides of an
inequality are multiplied or divided
by a negative number, the direction
of the inequality sign MUST BE
REVERSED.
Example 1:
-7(x – 2) - x < 4(5 – x) + 12
-7x + 14 - x < 20 - 4x + 12
(use distributive property)
- 8x + 14 < - 4x + 32
(simplify both sides)
- 8x + 4x + 14 < - 4x + 4x + 32
- 4x + 14 < 32
- 4x + 14 - 14 < 32 - 14
- 4x < 18
 18
x
4
Graph of solution ( -9
,)
2
(add 4x to both sides)
(simplify both sides)
(subtract 14 from both sides)
(simplify both sides)
(divide both sides by -4)
(
-9
2
9
x
(simplify)
2
Example 2:
 x  2 1  5x

 1
2
8
  x  2   1  5x 
8
  8
  8(1)
 2   8 
4( x  2)  1(1  5 x)  8
 4 x  8  1  5 x  8
x  7  8
x  15
(,15)
Example from today’s homework:
Something to think about:
• How would you graph the inequality 2 > x?
• What would this look like in interval notation?
Note that 2 > x is equivalent to x < 2.
Writing the inequality with the variable term on the
left makes it easier to “see” what the graph and
the interval notation should look like.
Interval notation: (-∞, 2)
This is an argument for working to put/keep your variables on the
left side of the expression as you solve linear inequalities.
Inequality Applications
Example: Six times a number, decreased by 2, is at
least 10. Find the number.
1.) UNDERSTAND
Let x = the unknown number.
“Six times a number” translates to 6x,
“decreased by 2” translates to 6x – 2,
“is at least 10” translates ≥ 10.
Example continued:
2.) TRANSLATE
Six times
a number
decreased
by 2
is at least
10
6x
–
2
≥
10
Example continued:
3.) SOLVE
6x – 2 ≥ 10
6x ≥ 12
x≥2
Add 2 to both sides.
Divide both sides by 6.
4.) INTERPRET
Check: Replace “number” in the original statement of the
problem with a number that is 2 or greater.
Six times 2, decreased by 2, is at least 10
6(2) – 2 ≥ 10
10 ≥ 10
State: The number is 2.
REMINDER:
In interval notation, ∞ and -∞
ALWAYS are enclosed by
a (round bracket)
NEVER by a [ square bracket].
The assignment on this material (HW 9) is due at the
start of the next class session.
Lab hours in 203:
Mondays through Thursdays
8:00 a.m. to 7:30 p.m.
Please remember to sign in!
Please open your laptops, log in to the MyMathLab
course web site, and open Daily Quiz 7.
• A scientific calculator may be used on this quiz. (No
graphing calculators, calculator apps or notes.)
• Write your name, date, and section info on the
worksheet handout and use this sheet for any
scratch work you do for this quiz.
• You have 5 minutes to finish this 2-question quiz.
• If you finish the quiz problems in less than 5
minutes, remember to check your work and your
online answers before submitting the quiz.
• After you submit your quiz, turn your worksheet in
to the TA, and then you are free to leave.