Transcript Do Now 12/7/06

Do Now 12/17/10

Copy HW in your planner.

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
Text p. 359, #4-14 even 22-36 even.
Text p. 366, #16-32 even, 36 & 40.
In your notebook, answer the following question.
Outside of the math classroom, where have you
heard phrases such as “at least” or “no more
than”? Give examples. How would you write
the phrases mathematically?
Chapter 6 Preview
“Solving and Graphing Linear Inequalities”
(6.1) Solve Inequalities Using Addition and Subtraction
(6.2) Solve Inequalities Using Multiplication and Division
(6.3) Solve Multi-Step Inequalities
(6.4) Solve Compound Inequalities
Winter break
(6.5) Solve Absolute Value Equations
(6.6) Solve Absolute Value Inequalities
(6.7) Graph Linear Inequalities in Two Variables
Section 6.1
“Solve Inequalities Using Addition and
Subtraction”
INEQUALITIES –
mathematical sentence formed by
placing a <, ≤, >, or ≥ between two
expressions.
11 - a ≤ 121
Writing Equations with Inequalities
Symbol
Meaning
Key phrases
=
Is equal to
The same as
<
Is less than
Fewer than
≤
Is less than or equal At most, no more
to
than
>
Is greater than
More than
≥
Is greater than or
equal to
At least, no less
than
 On
a number line, the GRAPH OF AN
INEQUALITY is the set of points that
represent ALL SOLUTIONS of the
inequality.
“Less than” and “greater
than” are represented
with an open circle.
Graph x < 8
5
6
“Less than or equal to”
and “greater than or equal
to” are represented with a
closed circle.
8
9
7
8
9
10
11
Graph x ≥ 11
10
11
12
13
14
15
The highest temperature recorded in the United States
was 134°F at Death Valley, California, in 1913. Use only
this fact to write and graph an inequality that
describes the temperatures in the United States.
SOLUTION
Let T represent a temperature (in degrees Fahrenheit)
in the United States. The value of T must be less than
or equal to 134. So, an inequality is T ≤ 134.
Write an inequality represented by the graph.
SOLUTION
The closed circle means that 8 is not a solution of
the inequality. Because the arrow points to the
left, all numbers less than 8 are solutions.
ANSWER
An inequality represented by the graph is x < 8.
Write an inequality represented by the graph.
SOLUTION
The closed circle means that – 2.5 is a solution of
the inequality. Because the arrow points to the
right, all numbers greater than – 2.5 are solutions.
ANSWER
An inequality represented by the graph is x > – 2.5.
Solving an Inequality…
Isolate the variable! Get ‘m’ by itself.
To get the ‘m’
by itself get rid of
“adding 4.”
Do the opposite.
“Subtract 4.”
m + 4 < 12
- 4 -4
Whatever you do to
one side of the Inequality
you must do the other side.
m<8
Solving an Inequality…
Isolate the variable! Get ‘n’ by itself.
To get the ‘n’
by itself get rid of
“subtracting 5.”
Do the opposite.
“Add 5.”
n-5≥ 6
+ 5 +5
Whatever you do to
one side of the inequality
you must do the other side.
n ≥ 11
Solve x – 5 > -3.5
Graph your solution
x – 5 > – 3.5
+5
+5
x > 1.5
Write original inequality.
Add 5 to each side.
Simplify.
ANSWER
The solutions are all real numbers greater
than 1.5. Check by substituting a number
greater than 1.5 for x in the original inequality.
Solve 9 ≥ x + 7
Graph your solution
9≥x+7
–7
–7
2≥x
Write original inequality.
Subtract 7 from each side.
Simplify.
ANSWER
You can rewrite 2 ≥ x as x ≥ 2.
The solutions are all real
numbers less than or equal
to 2.
Solve a real-world problem
LUGGAGE WEIGHTS
You are checking a bag at an airport. Bags can weigh
no more than 50 pounds. Your bag weighs 16.8 pounds.
Find the possible weights w (in pounds) that you can
add to the bag.
SOLUTION
Write a verbal model. Then write
and solve an inequality.
16.8
+
w
≤
50
Solve a real-world problem
16.8+ w < 50
16.8 + w – 16.8 < 50 – 16.8
w ≤ 33.2
ANSWER
You can add no
more than 33.2 pounds.
Write inequality.
Subtract 16.8 from each side.
Simplify.
Write and solve an inequality to find
the possible values of x.
 Perimeter
≤ 51.3 feet
14.2 + 15.5 + x ≤ 51.3
14.2 ft
29.7 + x ≤ 51.3
x ft
x ≤ 21.6
15.5 ft
Section 6.2
“Solve Inequalities Using Multiplication and
Division”
INEQUALITIES –
mathematical sentence formed by
placing a <, ≤, >, or ≥ between two
expressions.
11 - a ≤ 121
Solve 7 x  91
. Graph your solution
7x > 91
7
7
Write original inequality.
Divide each side by 7.
x > 13
Simplify.
Graph x > 13
10
11
12
13
14
15
16
x
Solve  5 Graph your solution.
4
x
< 5.
4
4 x <4 5
4
x < 20
Write original inequality.
Multiply each side by 4.
Simplify.
ANSWER
The solutions are all real numbers
less than 20. Check by substituting
a number less than 20 in the
original inequality.
Solve Inequalities When Multiplying and Dividing
by a NEGATIVE”
Multiplying and/or dividing each side
of an inequality by a NEGATIVE number
only produces an equivalent inequality
IF the inequality sign is REVERSED!!
m
 1.6 . Graph your solution.
Solve
7
m
– 7 < 1.6
m
– 7 – 7 > – 7 1.6
m > – 11.2
Write original inequality.
Multiply each side by – 7.
Reverse inequality symbol.
Simplify.
ANSWER
The solutions are all real numbers greater than – 11.2.
Check by substituting a number greater than – 11.2 in the
original inequality.
Solve  3x .24
–3x > 24.
–3x
< 24
–3
–3
x<–8
Write original inequality.
Divide each side by –3. Reverse
inequality symbol.
Simplify.
Solve a real-world problem
A student pilot plans to spend 80 hours on flight training to
earn a private license. The student has saved $6000 for training.
Which inequality can you use to find the possible hourly rates r
that the student can afford to pay for training?
A 80r < 6000 B 80r <
– 6000 C 80r >
– 6000 D 80r > 6000
SOLUTION
The total cost of training can be at most the amount of money
that the student has saved. Write a verbal model for the
situation. Then write an inequality.
80
r
The correct answer is B. A
<
–
B
6000
C
D
Solve a real-world problem
What are the possible
hourly rates that the
student can afford to pay
for training?
80
r ≤ 6000
80r ≤ 6000
80
80
r ≤ 75
Write inequality.
Divide each side by 80.
Simplify.
The student can afford to pay at most $75 per hour
for training.
Investigating Algebra Activity
“Inequalities with Negative Coefficients”
 Complete
the activity on page 362 in your
textbook. Read through the “Explore”
steps #1-3. Then complete the “Draw
Conclusions” problems #1-8.
24
Homework
 Text
p. 359, #4-14 even 22-36 even
 Text p. 366, #16-32 even, 36 & 40