Download Notes 6-1

Notes 6-1
Solving Inequalities:
Addition and Subtraction
y  2x – 3
I. Review: Inequalities
A. An inequality is a statement that two quantities are not
equal. The quantities are compared by using the following
signs:
<
A<B
A is less
than B.
>
≤
A>B
A≤B
A is greater
than B.
A is less
than or
equal to B.
≥
A≥B
A is greater
than or
equal to B.
≠
A≠B
A is not
equal to B.
A solution of an inequality is any value that makes the
inequality true. The set of all solutions of an inequality is its
solution set.
II. Graphing Inequalities on a Number Line
A. How to Graph Inequalities
Graphing
Inequalities
WORDS
ALGEBRA
All real numbers
less than 5
x<5
All real numbers
greater than -1
x > -1
All real numbers
less than or equal
to 1/2
x ≤ 1/2
All real numbers
greater than or
equal to 0
x≥0
GRAPH
–4
–4
–3 –2 –1
–3 –2
0
–1
1
2
1
0
3
2
4
3
5
4
6
5
6
1/2
–4
–4
–3 –2 –1
–3 –2 –1
0
0
1
1
2
2
3
3
4
4
5
5
6
6
B. Examples
Graph each inequality.
Draw an empty circle at 2.5.
Ex 1: c > 2.5
2.5
–4
–3 –2 –1
0
1
2
3
4
5
6
Shade in all the numbers greater
than 2.5 and draw an arrow pointing
to the right.
Graph each inequality.
Draw a solid circle at –3.
Ex 2:. m ≤ –3
−3
–8 –6 –4
–2
0
2
4
6
8
10 12
Shade in all numbers less
than –3 and draw an arrow
pointing to the left.
III. Writing Inequalities
Write the inequality shown by each graph.
Ex. 1.
x<2
Use any variable. The arrow points to the left, so use either < or ≤. The empty circle at 2
means that 2 is not a solution, so use <.
Ex. 2.
x ≥ –0.5
Use any variable. The arrow points to the right, so use either > or ≥. The solid circle at –
0.5 means that –0.5 is a solution, so use ≥.
IV. Set-Builder Notation
You have seen that one way to show the solution set of an inequality is by using a
graph. Another way is to use set-builder notation.
The set of all numbers x such that x has the given property.
{x : x < 6}
Read the above as “the set of all numbers x such that x is less than 6.”
V. Solving Inequalities
A. With a few exceptions (which we will get to tomorrow), we solve
inequalities in the same way we solve equations. Pretend the
inequality symbol is an equal symbol.
Ex 1: Solve the inequality and graph the solutions.
Since 12 is added to x,
subtract 12 from both sides
to undo the addition.
x + 12 < 20
–12 –12
x+0 < 8
x < 8
–10 –8 –6 –4 –2 0
2 4
6 8 10
Draw an empty circle at 8.
Shade all numbers less
than 8 and draw an arrow
pointing to the left.
B. Checking Solutions
Since there can be an infinite number of solutions to an inequality, it is not possible to
check all the solutions. You can check the endpoint and the direction of the inequality
symbol.
Ex: The solutions of x + 9 < 15 are given by x < 6.
Caution!
In Step 1, the endpoint should be a solution of the related equation, but it
may or may not be a solution of the inequality.
Ex 2: Solve the inequality and graph the solutions.
Since
5
is
subtracted
d – 5 > –7
from d, add 5 to both
+5 +5
sides to undo the
d + 0 > –2
subtraction.
d > –2
Draw an empty circle at –2.
–10 –8 –6 –4 –2 0 2 4 6 8 10
Shade all numbers
greater than –2 and
draw an arrow pointing
to the right.
Ex 3: Solve the inequality and graph the solutions.
0.9 ≥ n – 0.3
+0.3
+0.3
1.2 ≥ n – 0
1.2 ≥ n
0
1.2

1
Since 0.3 is subtracted
from n, add 0.3 to both
sides to undo the
subtraction.
Draw a solid circle at 1.2.
2
Shade all numbers
less than 1.2 and draw
an arrow pointing to
the left.
Ex 4: Solve each inequality and graph the solutions.
a. s + 1 ≤ 10
Since 1 is added to s, subtract 1 from
both sides to undo the addition.
–1 –1
s+0≤ 9
9
s ≤ 9
–10 –8 –6 –4 –2 0 2 4 6 8 10
b.
> –3 + t
+3 +3
> 0+t
t<
Since –3 is added to t, add 3 to both
sides to undo the addition.
–10 –8 –6 –4 –2 0
2 4
6 8 10
VI. Reading Math
Common Phrase
Equivalent Phrase
Symbol
No more than
Less than or equal to
≤
At most
Less than or equal to
≤
No less than
Greater than or equal to
≥
At least
Greater than or equal to
≥
Ex. 1. Ray’s dad told him not to turn on the air conditioner unless the temperature is at
least 85°F. Define a variable and write an inequality for the temperatures at which Ray
can turn on the air conditioner. Graph the solutions.
Let t represent the temperatures at which Ray can turn on the air conditioner.
Turn on the AC when temperature
is at least
t
≥
85
Draw a solid circle at 85. Shade all
numbers greater than 85 and draw an
arrow pointing to the right.
t  85
70
85°F
75
80
85
90
Ex. 2: A store’s employees earn at least $8.25 per hour. Define a variable and
write an inequality for the amount the employees may earn per hour. Graph the
solutions.
Let d represent the amount an employee can earn per hour.
An employee earns
at least
$8.25
≥
d
d ≥ 8.25
8.25
−2
0
2
4
6
8 10 12 14 16 18
8.25
4. A certain restaurant has room for 120
customers. On one night, there are 72
customers dining. Write and solve an
inequality to show how many more people
can eat at the restaurant.
x + 72 ≤ 120; x ≤ 48, where x is a natural
number
Lesson Quiz: Part I
1. Describe the solutions of 7 < x + 4.
all real numbers greater than 3
2. Graph h ≥ –4.75
–5
–4.75
–4.5
Write the inequality shown by each graph.
3.
4.
x≥3
x < –5.5
Lesson Quiz: Part II
5. A cell phone plan offers free minutes for no
more than 250 minutes per month. Define a
variable and write an inequality for the possible
number of free minutes. Graph the solution.
Let m = number of minutes.
0 ≤ m ≤ 250
0
250
Lesson Quiz: Part III
Solve each inequality and graph the solutions.
1. 13 < x + 7
x>6
2. –6 + h ≥ 15
h ≥ 21
3. 6.7 + y ≤ –2.1
y ≤ –8.8