Transcript Inequalities - Hale`s Math Minions

Unit III
Inequalities
2.6
Solving Linear Inequalities
1. Represent solutions to inequalities
graphically and using set notation.
2. Solve linear inequalities.
Inequalities work like equations, but they tell you whether one expression is
bigger or smaller than the expression on the other side.
An inequality is like an equation, but instead of an
equal sign (=) it has one of these signs:
An inequality is a mathematical sentence that states that two expressions are not equal.
"Solving'' an inequality means finding all of its solutions. A "solution'' of an inequality is a
number which when substituted for the variable makes the inequality a true statement.
Inequality symbols
<
•less than
•fewer than
>

•greater than •less than or
equal to
•more than
•no more
•exceeds
than
•at most

•greater than
or equal to
•no less than
•at least
Graphing Inequalities
x<c
When x is less than a constant, you darken in the part of the number line that is to the left of the
constant. Also, because there is no equal line, we are not including where x is equal to the constant. That
means we are not including the endpoint. One way to notate that is to use an open hole at that point.
x>c
When x is greater than a constant, you darken in the part of the number line that is to the right of the
constant. Also, because there is no equal line, we are not including where x is equal to the constant. That
means we are not including the endpoint. One way to notate that is to use an open hole at that point.
x<c
When x is less than or equal to a constant, you darken in the part of the number line that is to the left of
the constant. Also, because there is an equal line, we are including where x is equal to the constant. That
means we are including the endpoint. One way to notate that is to use an closed hole at that point.
x>c
When x is greater than or equal to a constant, you darken in the part of the number line that is to the right
of the constant. Also, because there is an equal line, we are including where x is equal to the
constant. That means we are including the endpoint. One way to notate that is to use a closed hole at
that point.
Graphing Inequalities.
Graph each of these inequalities.
8. State the inequality represented
on the number line below.
k > –7
Applications
Teresa is only allowed to swim outside if the temperature outside is at least 85 °F.
Write an inequality that shows the temperature in degrees Fahrenheit at which
Teresa is allowed to swim.
In order to achieve an ‘A’ in math, Ivy needs to score more than 95% on her next
test. Write an inequality that shows the test score Ivy needs to achieve in
order to earn her ‘A’ in math.
i > 95
Inequalities – Interval Notation
[( smallest, largest )]
Parentheses: endpoint is not included <, >
Bracket: endpoint is included ≤, ≥
Infinity: always uses a parenthesis
x<2
( –∞, 2)
x≥2
[2, ∞)
4<x<9
3-part inequality
(4, 9)
Inequalities – Set-builder Notation
{variable | condition }
pipe
{ x | x  5}
The set of all x such that x is greater than or equal to 5.
x<2
( –∞, 2)
{x|x<2}
[2, ∞)
{ x | x ≥ 2}
(4, 9)
{ x | 4 < x < 9}
x≥2
4<x<9
Inequalities
Graph, then write interval notation and set-builder notation.
x≥5
[
Interval Notation: [ 5, ∞)
Set-builder Notation: { x | x ≥ 5}
x < –3
)
Interval Notation: (– ∞, –3)
Set-builder Notation: { x | x < –3 }
Inequalities
Graph, then write interval notation and set-builder notation.
1<a<6
(
)
Interval Notation: ( 1, 6 )
Set-builder Notation: { a | 1 < a < 6 }
–7 < x ≤ 3
(
]
Interval Notation: (– 7, 3]
Set-builder Notation: { x | –7 < x ≤ 3 }
Addition/Subtraction Property for Inequalities
If a < b, then a + c < b + c
If a < b, then a - c < b – c
In other words, adding or subtracting the same expression to both sides of an inequality does not
change the inequality.
Ex. A Solve and graph the solution of x – 2 > 5
on a number line.
Solve:
x–2>5
x–2>5
+2
+2
Solve:
Using addition
property of
inequalities
Subtraction
property of
inequalities
x>7
Graph:
Graph:
(
–7
0
7
–1 0 1 2 3 4 5 6
Practice
Multiplication/Division Properties for Inequalities
when multiplying/dividing by a positive value If a < b AND c is positive, then ac < bc
If a < b AND c is positive, then a/c < b/c
In other words, multiplying or dividing the same POSITIVE number to both sides of an inequality does
not change the inequality.
Solve 1
5
Solve 3 x  12
x2
Solution
1
x2
5
(5)
1
x  2 (5)
5
x > 10
Solution
Given Inequality
3 x  12
___ ___
3
3
Multiplication Property
x<4
Given Inequality
Division Property
Multiplication/Division Properties for Inequalities with NEGATIVE Numbers
Given real numbers a, b, and c, if a > b and c < 0 then ac < bc.
a < b
Given real numbers a, b, and c, if a > b and c < 0 then
c
c
In other words, multiplying or dividing the same NEGATIVE number to both sides of an inequality
REVERSES the direction of the inequality, otherwise the inequality statement will be false.
Solve:  1 x  2
Solve: 13 y  39
2
1
2
Solution:  x  2
1
( 2)  x  2( 2)
2
Solution: 13 y  39
13 y
39

13
13
x < –4
Remember — the sign
needs to change.
Remember — the sign
needs to change.
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!
Practice
9x  36
2
y  10
5
6 y  48
1
g 7
7
72x  8
x4
y  25
y 8
g  49
y
1
9
x>4
x>7
Applications
Laura has $5.30 to spend on her lunch. She wants to buy a chicken salad costing
$4.20 and decides to spend the rest on fruit. Each piece of fruit costs 45¢.
Write an inequality to represent this situation, and then solve it to find how many
pieces of fruit Laura can buy.
Laura can buy 2 pieces of fruit
Audrey is selling magazine subscriptions to raise money for the school library. The library
will get $2.50 for every magazine subscription she sells. Audrey wants to raise at least $250
for the library.
Write and solve an inequality to represent the number of magazine subscriptions, x, Audrey
needs to sell to reach her goal.
Audrey must sell at least 100
magazine subscriptions
Multistep Inequalities
To simplify and therefore solve an inequality in one variable such as x, you need to
isolate the terms in x on one side and isolate the numbers on the other.
Solving Inequalities
1. Multiply out any parentheses
2. Simplify each side of the inequality.
3. Remove number terms from one side
4. Remove x-terms from the other side.
5. Multiply or divide to get an x-coefficient of 1
Practice
1)
2)
5x + 1 > 3(x + 3)
x>4
3)
4(x + 1)
> 2x
6
x < 0.5
7(a – 4)
2
a > -28
4)
5)
Solving Inequalities
If we multiply (or divide) by a negative, reverse
the direction of the inequality!!!!!
4 x  16
4
4
x4
 4 x  16
4
4
x  4
4 x  16
4
4
x  4
Solving Inequalities
Solve then graph the solution and write it in interval notation
and set-builder notation.
3x  4  7
4 4
Don’t write = !
3x  3
3
(
3
x 1
Interval Notation: ( 1, ∞ )
Set-builder Notation: { x | x > 1 }
Solving Inequalities
Solve then graph the solution and write it in interval notation
and set-builder notation.
4  9k  4k  19
 4k
 4k
4  5k  19
4
4
 5k  15
5 5
]
k  3
Interval Notation: (– ∞, –3 ]
Set-builder Notation: { k | k ≤ –3 }
Solving Inequalities
Solve then graph the solution and write it in interval notation
and set-builder notation.
5
 p  10
3
5 

3
p   3 10
 3 
 5p  30
5
5
)
p6
Interval Notation: (– ∞, 6 )
Set-builder Notation: { p | p < 6 }
Solving Inequalities
Solve then graph the solution and write it in interval notation
Moving variable to the right.
and set-builder notation.
12m  14  15m  5
1
1
6m  7  3m  1
 12m
 12m
5
2
 14  3m  5
1
1


10 6m  7  10 3m  1
5
5
 5
2
 9  3m
26m  7  53m  1
3
12m  14  15m  5
 15m
 15m
 3m  14  5
 14
 3m  9
3
3m
3
3
m  3
[
 14
Interval Notation: [– 3, ∞ )
Set-builder Notation: { m | m ≥ – 3 }
Compound Inequalities
A compound inequality is two inequalities together
— for example, 2x + 1 < 5 and 2x + 1 > –1.
The word “and” means the compound
inequality below is a “conjunction.”
2x + 1 < 5 and 2x + 1 > –1
Solve and graph the inequality
–1 < 2x + 1 < 5.
The goal is to get x by itself.
–1 < 2x + 1 < 5
-1
You can rewrite a conjunction as a single
mathematical statement, usually involving two
inequality signs, like this:
–1 < 2x + 1 < 5
* The solution to a conjunction must
satisfy both inequalities — both
inequalities must be true.
-1
-1
–2 < 2x < 4
–1 < x < 2
Subtract 1
Divide by 2 to get x in
the middle
the solution is any number greater than –1 but
less than 2
Disjunction Problems Include the Word “Or”
Here’s an example of a disjunction:
3x – 4 < –4 or 3x –4 > 4
The solution to a disjunction is all the numbers that satisfy either one inequality or
the other.
Solve and graph the solution set of
3x – 4 < –4 or 3x – 4 > 4.
3x – 4 + 4 < –4
3x – 4 + 4 < –4 + 4
or
or
3x – 4 + 4 > 4
3x – 4 + 4 > 4 + 4
3x < 0
or
3x > 8
x<0
or
8
x>
3
Add 4
Divide by 3
Compound Inequalities Practice
1) –11 < –4g + 5 < –3
–16 < –4g < –8
2<g<4
2)
8c – 4 > 92 or 8c – 4 < –12
3)
2y + 2 < 4y – 4 or 4y – 4 > 5y + 2
4)
–9g – 7 > 2 or –9g – 7 > 20
8c > 96 or 8c < –8
c > 12 or c < –1
y > 3 or y < –6
5)
6)
–11 <
c–9
7
<5
–77 < c – 9 < 35
–68 < c < 44
The sum of three consecutive even integers is between 82 and 85. Find the
numbers.
26, 28, and 30
The formula C =
5
9
(F – 32) is used to convert degrees Fahrenheit
to degrees Celsius.
The temperature inside a greenhouse falls to a minimum of 65 °F at night and
rises to a maximum of 120 °F during the day. Find the corresponding
temperature range in degrees Celsius.
18 °C – 49 °C
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
Graphing Linear Inequalities
in Two Variables
•SWBAT graph a linear
inequality in two variables
•SWBAT Model a real life
situation with a linear
inequality.
Some Helpful Hints
•If the sign is > or < the line is
dashed
•If the sign is  or  the line will be
solid
When dealing with just x and y.
•If the sign > or  the shading
either goes up or to the right
•If the sign is < or  the shading
either goes down or to the left
Graphing an Inequality in Two Variables
Graph x < 2
Step 1: Start by graphing
the line x = 2
Now what points
would give you less
than 2?
Since it has to be x < 2
we shade everything to
the left of the line.
Graphing a Linear Inequality
Sketch a graph of y  3
Using What We Know
Sketch a graph of x + y < 3
Step 1: Put into
slope intercept form
y < -x + 3
Step 2: Graph the
line y = -x + 3