Transcript Graphing and Writing inequalities ppt

Graphing and Writing
Inequalities
with One Variable
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of contents
When you need to use an inequality to solve a word
problem, you may encounter one of the phrases below.
Equivalent
Translation
Important
Words
Sample
Sentence
is more than
Trenton is more
than 10 miles
away.
t > 10
is greater than
A is greater than
B.
A>B
must exceed
The speed must
exceed 25 mph.
The speed is
greater than 25
mph.
s > 25
When you need to use an inequality to solve a word problem, you may
encounter one of the phrases below.
Important
Words
Sample
Sentence
Equivalent
Translation
cannot exceed
Time cannot
exceed 60
minutes.
Time must be
less than or
equal to 60
minutes.
t < 60
is at most
At most, 7
students were
late for class.
Seven or fewer
students were
late for class.
n<7
is at least
Bob is at least
14 years old.
Bob's age is
greater than or
equal to 14.
B > 14
How are these inequalities read?
2+2>3
Two plus two is greater than 3
2+2>3
Two plus two is greater than or equal to 3
2+2≥4
Two plus two is greater than or equal to 4
2+2<5
Two plus two is less than 5
2+2≤5
2+2≤4
Two plus two is less than or equal to 5
Two plus two is less than or equal to 4
Writing inequalities
Let's translate each statement into an inequality.
words
x is less than 10
translate to
x
<
10
inequality statement
20 is greater than or equal to
y
20
y
>
You try a few:
1. 14 is greater than a
2. b is less than or equal to 8
3. 6 is less than the product of f and 20
4. The sum of t and 9 is greater than or equal to 36
5. 7 more than w is less than or equal to 10
6. 19 decreased by p is greater than or equal to 2
7. Fewer than 12 items
8. No more than 50 students
9. At least 275 people attended the play
Do you speak math?
Change the following expressions from English into
math.
Double a number is at most four.
2xAnswer
≤4
Three plus a number is at least six.
3+x≥6
Answer
Five less than a number is less than twice that number.
Answer
x-5<
2x
The sum of two consecutive numbers is at least
thirteen.
x + (xAnswer
+ 1) ≥ 13
Three times a number plus seven is at least nine.
3x + 7 > 9
Answer
A store's employees earn at least $7.50 per hour. Define
a variable and write an inequality for the amount the
employees may earn per hour.
Let e represent an employee's wages.
An employee earns
at least
$7.50
e
>
7.5
7.5
0
1
2
3
4
5
6
7
8
9
10
Try this:
The speed limit on a road is 55 miles per hour. Define a
variable and write an inequality.
118
You have $200 to spend on clothes. You
already spent $140 and shirts cost $12.
Which equation shows this scenario?
A
B
C
D
200 < 12x + 140
200 ≤ 12x + 140
200 > 12x + 140
200 ≥ 12x + 140
119 A sea turtle can live up to 125 years. If one is
already 37 years old, which scenario shows
how many more years could it live?
A
B
C
125 < 37 + x
125 ≤ 37 + x
125 > 37 + x
D
125 ≥ 37 + x
120
The width of a rectangle is 3 in longer than
the length. The perimeter is no less than 25
inches.
A
B
C
D
4a + 6 < 25
4a + 6 ≤ 25
4a + 6 > 25
4a + 6 ≥ 25
121 The absolute value of the sum of two
numbers is less than or equal to the sum of
the absolute values of the same two numbers.
A
B
C
D
Solution Sets
A solution to an inequality is NOT a single
number. It will have more than one value.
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
This would be read as the solution set is all
numbers greater than or equal to negative 5.
Let's name the numbers that are solutions of the given
inequality.
r > 10
Which of the following are solutions? {5, 10, 15, 20}
5 > 10 is not true
So, not a solution
10 > 10 is not true
So, not a solution
15 > 10 is true
So, 15 is a solution
20 > 10 is true
So, 20 is a solution
Answer:
{15, 20} are solutions of the inequality r > 10
Let's try another one.
30 ≥ 4d; {3, 4, 5, 6, 7, 8}
30 ≥ 4d
30 ≥ (4)3
reveal
30click≥to12
30 ≥ 4d
30 ≥ (4)4
reveal
30click
≥ to16
30 ≥ 4d
30 ≥ (4)5
reveal
30click
≥ to20
30 ≥ 4d
30 ≥ (4) 6
30click≥ to24
reveal
30 ≥ 4d
30 ≥ (4)7
30click
≥ to28
reveal
30 ≥ 4d
30 ≥ (4)8
30click
≥ to32
reveal
Graphing Inequalities - The Circle
An open circle on a number shows that the number
is not part of the solution. It is used with "greater
than" and "less than". The word equal is not
included.< >
A closed circle on a number shows that the number
is part of the solution. It is used with "greater than
or equal to" and "less than or equal to". < >
Graphing Inequalities - The Arrow
The arrow should always point in the direction of those
numbers who satisfy the inequality.
*If the variable is on the left side of the inequality, then
< and ≤ will show an arrow pointing left.
*If the variable is on the left side of the inequality, then
> and ≥ will show an arrow pointing right.
Notice that < and ≤ look like an arrow pointing left
and that > and ≥ look like an arrow pointing right.
But what if the variable isn't on the left?
Do the opposite of where the inequality symbol points.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Graphing Inequalities
What is the number in the inequality?
What kind of circle should be used?
In what direction does the line go?
Graphing Inequalities
x is less than 5
Step 1: Rewrite this as x < 5.
Step 2: What kind of circle? Because it is less than, it
does not include the number 5 and so it is an open
circle.
-5
-4
-3
-2
-1
0
1
2
3
4
5
x<5
Step 3: Draw an arrow on the number line showing all
possible solutions. Numbers greater than the variable, go
to the right. Numbers less than the variable, go to the left.
-5
-4
-3
-2
-1
1
0
2
3
5
4
Step 4: Draw a line, thicker than the horizontal line, from the
dot to the arrow. This represents all of the numbers that
fulfill the inequality.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Graphing Inequalities
x is less than or equal to 5
Step 1: Rewrite this as x ≤ 5.
Step 2: What kind of circle? Because it is less than or
equal to, it does include the number 5 and so it is a
closed circle.
-5
-4
-3
-2
-1
0
1
2
3
4
5
x≤5
Step 3: Draw an arrow on the number line showing all
possible solutions. Numbers greater than the variable, go
to the right. Numbers less than the variable, go to the left.
-5
-4
-3
-2
-1
1
0
2
3
5
4
Step 4: Draw a line, thicker than the horizontal line, from the
dot to the arrow. This represents all of the numbers that
fulfill the inequality.
-5
-4
-3
-2
-1
0
1
2
3
4
5
You try
Graph the inequality
x>2
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
.
0
5
.
02
5
3
4
5
6
7
8
9
10
click 2 on the
number line for
answer
Graph the inequality
-3 > x
-10 -9
-8
-7
-6
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
click -3 on the
number line for
answer
Try these.
Graph the inequalities.
1. x > -3
-5
-4
.
0
5
-3
.
-2
-1
0
1
2
3
4
5
-3
-2
-1
0
1
2
3
4
5
2. x < 4
-5
-4
Try these.
State the inequality shown.
1.
-5
-4
-3
-2
-1
0
1
2
3
4
5
2.
-5
-4
-3
-2
-1
0
1
2
3
4
5
122 This solution set would be x > -4.
A
B
-10 -9
-8
-7
-6
True
False
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
123 State the inequality shown.
-5
-4
-3
-2
-1
A
x>3
B
x<3
C
x<3
D
x>3
0
1
2
3
4
5
124
State the inequality shown.
5
6
7
8
9
A
11 < x
B
11 > x
C
11 > x
D
11 < x
10
11 12 13
14 15
125
State the inequality shown.
-5
-4
-3
-2
-1
A
x > -1
B
x < -1
C
x < -1
D
x > -1
0
1
2
3
4
5
126
State the inequality shown.
-5
-4
-3
-2
-1
A
-4 < x
B
-4 > x
C
-4 < x
D
-4 > x
0
1
2
3
4
5
127
State the inequality shown.
-5
-4
-3
-2
A
x>0
B
x<0
C
x<0
D
x>0
-1
0
1
2
3
4
5