Transcript Linear inequalities 1
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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 ≥ 89 ≥ 86 ≥ 83 ≥ 80 ≥ 75 ≥ 70 ≥ 67 < 67
Why you should keep taking the practice quiz until you can score at least 90%: (A) (B+) (low F) (almost a C-)
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 12 = 7 – 3
y x
> 3 12 ≤ 7 – 3
y
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 :
-4 ( 7 ]
∞
The inequality x 7 is expressed in
interval notation
as ( , 7]
∞
The inequality x > -4 is 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}
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) 5)
6)
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).
Isolate variable by dividing both sides by the number in front of the variable (multiplication property of inequality).
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 -7
x
+ 14 -
x
< 20 - 4
x
+ 12 (use distributive property) - 8
x
+ 14 < - 4
x
+ 32 - 8
x
+ 4
x
+ 14 < - 4
x
+ 4
x
+ 32 - 4
x
+ 14 < 32 (simplify both sides) (add 4x to both sides) (simplify both sides) - 4
x
+ 14 - 14 < 32 - 14 4
x x
< 18 18 4 (subtract 14 from both sides) (simplify both sides) (divide both sides by -4)
x
9 2 (simplify) -9 Graph of solution ( , ) 2 2 -9 (
Example 2:
x
2 1 2 8 4 (
x x
2 2 2 ) 8 1 5
x
8 ( 1 8 5 5
x
) 1
x
8 8 ( 1 ) 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 6
x
, “decreased by 2” translates to 6
x
– 2, “is at least 10” translates ≥ 10.
Example continued:
2.) TRANSLATE
Six times a number decreased by 2 is at least 10 6
x
– 2 ≥ 10
Example continued: 3.) SOLVE
6
x
– 2 ≥ 10 6
x
≥ 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.
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