Transcript Solving an Inequality…

Do Now 12/21/11
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Copy HW in your planner.
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Cumulative Test Chapter 1-5 tomorrow
Text p. 359, #4-14 even 22-36 even.
Text p. 366, #16-32 even, 36 & 40.
Text p. 372, #10-24 even, 30-34 even, 38, 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
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
m<8
Whatever you do to
one side of the Inequality
you must do the other side.
5
6
7
8
9
10
11
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
n ≥ 11
Whatever you do to
one side of the inequality
you must do the other side.
8
9
10
11
12
13
14
15
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 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.
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
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.
Section 6.3
“Solve Multi-Step Inequalities”
The steps for solving two-step and multi-step
equations can be applied to linear inequalities.
Solving Multi-Step Inequalities
STEP 1STEP 2STEP 3STEP 4STEP 5STEP 6-
Use distributive property and combine like terms.
Collect variables on one side of the inequality.
“Undo” addition and/or subtraction.
“Undo” multiplication and/or division.
Solve for the variable.
Check your work.
REMEMBER!!!!!
Multiplying and/or dividing each side
of an inequality by a NEGATIVE number
only produces an equivalent inequality
IF the inequality sign is REVERSED!!
Solve
Solve 3x
3x–– 77<<8.8.Graph
Graphyour
your solution.
solution.
3x – 7 < 8
3x < 15
x<5
Write original inequality.
Add 7 to each side.
Divide each side by 3.
ANSWER
The solutions are all real numbers less than 5. Check
by substituting a number less than 5 in the original
inequality.
Solve – 0.6(x – 5) <
– 15
–0.6(x – 5) <– 15
–0.6x + 3 <
– 15
– 0.6x
–< 12
x –> –20
Write original inequality.
Distributive property
Subtract 3 from each side.
Divide each side by – 0.6. Reverse
inequality symbol.
ANSWER
The solutions are all real numbers greater than equal
-20. Check by substituting a number more than -20 in
the original inequality.
Solve 6x – 7 > 2x+17. Graph your solution.
6x – 7 > 2x+17
Write original inequality.
6x > 2x+24
Add 7 to each side.
4x > 24
Subtract 2x from each side.
x>6
Divide each side by 4.
ANSWER
The solutions are all real numbers greater than 6.
Solve:
14x + 5 < 7(2x – 3)
14x + 5 < 7(2x – 3)
Write original inequality.
14x + 5 < 14x – 21
Distributive property
5 < – 21
Subtract 14x from each side.
There are no solutions because 5 < – 21 is false.
**HINT**
If an inequality is equivalent to an inequality that is false,
such as 5 < -21, then the solution of the inequality has
NO SOLUTION.
12x – 1 > 6(2x – 1)
12x – 1 > 6(2x – 1)
Write original inequality.
12x – 1 > 12x – 6
Distributive property
–1>–6
Subtract 12x from each side.
All real numbers are solutions because – 1 > – 6 is true.
**HINT**
If an inequality is equivalent to an inequality that is true,
such as -1 > -6, then the solutions of the inequality are
ALL REAL NUMBERS .
Graphs of “No Solution” and
“All Real Numbers”
 “No
Solution”
 “All
Real Numbers”
Car Wash
Use the sign shown. A gas station charges $.10
less per gallon of gasoline if a customer also
gets a car wash. What are the possible amounts
(in gallons) of gasoline that you can buy if you
also get a car wash and can spend at most $20?
Because you are getting a car wash, you will
pay $2.09 – 2 $.10 = $1.99 per gallon of gasoline.
Let g be the amount (in gallons) of gasoline
that you buy.
STEP 1
Write a verbal model. Then write an inequality.
Price of
gasoline
(dollars/gallon)
1.99
•
Amount of
gasoline
(gallons)
g
+
+
Price of
car wash
(dollars)
8
<
<
–
Maximum
amount
(dollars)
20
STEP 2
Solve the inequality.
1.99g + 8 ≤ 20
1.99g ≤ 12
g ≤ 6.03015. . .
Write inequality.
Subtract 8 from each side.
Divide each side by 1.99.
You can buy up to slightly more than 6 gallons of gasoline.
CHECK
You can use a table to check the
reasonableness of your
answer.The table shows that you
will pay $19.94 for exactly 6 gallons
of gasoline. Because $19.94 is less
than $20, it is reasonable to
conclude that you can buy slightly
more than 6 gallons of gasoline.
Gasoline
(gal)
Total amount spent
(dollars)
0
8.00
1
9.99
2
11.98
3
13.97
4
15.96
5
17.95
6
19.94
24
Homework
 Text
p. 359, #4-14 even 22-36 even
 Text p. 366, #16-32 even, 36 & 40
 Text p. 372, #10-24 even, 30-34 even, 38, 40