Transcript PPT Review Chapter 2 Inequalities

CHAPTER 6 REVIEW
Solving and
Graphing
Inequalities
An inequality is like an equation,
but instead of an equal sign (=) it
has one of these signs:
< : less than
≤ : less than or equal to
> : greater than
≥ : greater than or equal to
“x < 5”
means that whatever value x
has, it must be less than 5.
Try to name ten numbers that
are less than 5!
Numbers less than 5 are to the left
of 5 on the number line.
-25 -20 -15 -10 -5
0
5
10 15 20 25
• If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.
• There are also numbers in between the integers, like
2.5, 1/2, -7.9, etc.
• The number 5 would not be a correct answer,
though, because 5 is not less than 5.
“x ≥ -2”
means that whatever value x
has, it must be greater than or
equal to -2.
Try to name ten numbers that
are greater than or equal to 2!
Numbers greater than -2 are to the
right of 5 on the number line.
-25 -20 -15 -10 -5
0
5
10 15 20 25
-2
• If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.
• There are also numbers in between the integers,
like -1/2, 0.2, 3.1, 5.5, etc.
• The number -2 would also be a correct answer,
because of the phrase, “or equal to”.
Where is -1.5 on the number line?
Is it greater or less than -2?
-25 -20 -15 -10 -5
0
5
10 15 20 25
-2
• -1.5 is between -1 and -2.
• -1 is to the right of -2.
• So -1.5 is also to the right of -2.
Solve an Inequality
w+5<8
w + 5 + (-5) < 8 + (-5)
w<3
All numbers less
than 3 are
solutions to this
problem!
More Examples
8 + r ≥ -2
8 + r + (-8) ≥ -2 + (-8)
r ≥ -10
All numbers from -10 and up (including
-10) make this problem true!
More Examples
x - 2 > -2
x + (-2) + (2) > -2 + (2)
x>0
All numbers greater than 0 make this
problem true!
More Examples
4+y≤1
4 + y + (-4) ≤ 1 + (-4)
y ≤ -3
All numbers from -3 down (including -3)
make this problem true!
There is one special case.
● Sometimes you may have to reverse the
direction of the inequality sign!!
● That only happens when you
multiply or divide both sides of the
inequality by a negative number.
Example:
Solve: -3y + 5 >23
●Subtract 5 from each side.
-5 -5
-3y > 18
-3
-3 ●Divide each side by negative 3.
y < -6 ●Reverse the inequality sign.
●Graph the solution.
-6
0
Try these:
1.) Solve 2x + 3 > x + 5
-x
-x
2.)Solve - c – 11 >23
+ 11
x+3>5
-3 -3
-c > 34
-1 -1
x>2
c < -34
3.) Solve 3(r - 2) < 2r + 4
3r – 6 < 2r + 4
-2r
-2r
r–6<4
+6 +6
r < 10
+ 11
You did
to reverse
. . remember
.Good
didn’tjob!
you?
the signs . . .
 15  4 x  7  5
7
7 7
 8  4 x  12
4 4 4
2  x 3
Example:
2x  6  4x  8
- 4x
- 4x
 2x  6  8
+ 6 +6
 2 x  14
-2
We turned the sign!
-2
x  7
Ring the alarm!
We divided by a
negative!
Remember Absolute Value
Ex: Solve 6x-3 = 15
6x-3 = 15 or 6x-3 = -15
6x = 18 or 6x = -12
x = 3 or x = -2
* Plug in answers to check your solutions!
Ex: Solve 2x + 7 -3 = 8
Get the abs. value part by itself first!
2x+7 = 11
Now split into 2 parts.
2x+7 = 11 or 2x+7 = -11
2x = 4 or 2x = -18
x = 2 or x = -9
Check the solutions.
Ex: Solve & graph.
4 x  9  21
• Becomes an “and” problem
15
3 x 
2
-3
7
8
Solve & graph.
3x  2  3  11
• Get absolute value by itself first.
3x  2  8
• Becomes an “or” problem
3x  2  8 or 3x  2  8
3x  10 or
3x  6
10
x
or x  2
3
-2
3
4
Example 1:
This is an ‘or’ statement.
(Greator). Rewrite.
● |2x + 1| > 7
● 2x + 1 > 7 or 2x + 1 >7
● 2x + 1 >7 or 2x + 1 <-7
●
x > 3 or
In the 2nd inequality, reverse the
inequality sign and negate the
right side value.
Solve each inequality.
x < -4
Graph the solution.
-4
3
Example 2:
● |x -5|< 3
This is an ‘and’ statement.
(Less thand).
● x -5< 3 and x -5< 3
● x -5< 3 and x -5> -3
●
●
Rewrite.
In the 2nd inequality, reverse the
inequality sign and negate the
right side value.
x < 8 and x > 2
2<x<8
Solve each inequality.
Graph the solution.
2
8
Absolute Value Inequalities
Case 1 Example:
x  3  5
x  2
x 3  5
and
x 3  5
x8
2  x  8
Absolute Value Inequalities
Case 2 Example: 2 x  1  9
2x  1   9
2x   10
x  5
or
2x  1  9
2x  8
x4
x   5 OR x  4
Absolute Value
• Answer is always positive
• Therefore the following examples
cannot happen. . .
3x  5  9
Solutions: No solution
Sean earns $6.70 per hour working after school. He needs
at least $155 (≥ 155) for a stereo system. Write an
inequality that describes how many hours he must work to
reach his goal.
To figure your pay check:
Per means to multiply
You multiply rate times hours worked
$6.70 h ≥ 155
$6.70
$6.70
h  23.1343
h  24
So 23 hours
is not enough
So Sean must
Work 24 hours
To earn enough
Money.
Sean earns $6.85 per hour working after school. He needs
at least $310 (≥ 310) for a stereo system. Write an
inequality that describes how many hours he must work to
reach his goal.
To figure your pay check:
Per means to multiply
You multiply rate times hours worked
$6.85 h ≥ 310
$6.85
$6.85
h  45.25547
h  46
So 45 hours
is not enough
So Sean must
Work 46 hours
To earn enough
Money.
The width of a rectangle is 11 cm. Find all possible
values for the length of the rectangle if the perimeter is
at least 300 cm. ( ≥ 300 cm)
• 2L + 2W = p
2 L  2(11)  300
2 L  22  300
22 22
2 L  278
2
2
L  139
The width of a rectangle is 31 cm. Find all possible
values for the length of the rectangle if the perimeter is
at least 696 cm. ( ≥ 696 cm)
• 2L + 2W = p
2 L  2(31)  696
2 L  62  696
62 62
2 L  634
2
2
L  317
The perimeter of a square is to be between 11 and 76 feet,
inclusively. Find all possible values for the length of its sides.
11  p  76
11  4s  76
4
4
4
2.75  s  19
• 4 equal sides
• Perimeter – add up
sides s
s
s
s
P = s +s+s+s=4s
The perimeter of a square is to be between 14 and 72 feet,
inclusively. Find all possible values for the length of its sides.
14  p  72
14  4s  72
4
4
4
3.5  s  18
• 4 equal sides
• Perimeter – add up
sides s
s
s
s
P = s +s+s+s=4s
Three times the difference of a number and 12 is at most
87. Let x represent the number and find all possible
values for the number.
3 ( x  12)  87
3x 36  87
36 36
3x  123
3
3
x  41
Marlee rented a paddleboat at the Park for a fixed charge of
$2.50 plus $1.50 per hour. She wants to stay out on the water
as long as possible. How many hours can she use the boat
without spending more than $7.00?
$2.50  $1.50h  $7.00
$2.50
$2.50
$1.50h  $4.50
$1.50
$1.50
h3