6_5 Graphing Linear Inequalities

```6.5 Slope intercept form
for Inequalities:
Linear Inequality: is a linear equation with
an inequality sign (< , ≤, >, ≥)
Solution of an Inequality: is an ordered
pair (x, y) that makes the inequality true.
GOAL:
Whenever we are given a graph we must be able
to provide the equation of the function.
Slope-Intercept Form: The linear equation of a
nonvertical line with an inequality sign:
y (<, ≤, >, ≥) m x + b
Slope =
−
−
=
⍙
=
⍙
y-intercept
y crossing
http://mathgraph.idwvogt.com/examples.html
Whenever we are given a graph we must be able
to provide the equation of the function.
y<mx+b
dash line
Whenever we are given a graph we must be able
to provide the equation of the function.
y>mx+b
dash line
Whenever we are given a graph we must be able
to provide the equation of the function.
y≤mx+b
Solid line
Whenever we are given a graph we must be able
to provide the equation of the function.
y≥mx+b
Solid line
EX:
Provide the equation of the inequality.
Solution: Since line is dashed and shaded at
the bottom we use <. Also, the inequality
must be in
slope-intercept form:
Y < mx + b
1. Find the y-intercept
In this graph b = +1.
2. Find another point
to get the slope.
A(0,5)
A(0,1)
B(3,-2)
Use the equation of slope to find the slope:
−
−
=
−−
−
=
−

= -1
The slope-intercept
form inequality is:
A(0,1)
B(3,-2)
y < -1x + 1
Remember:
This means that if you start a 1 and move down
one and over to the right one, and continue this
pattern. We shade the bottom since it is <.
When work does not need to be shown:
(EOC Test)
look at the triangle
points.
Count the number
of square going up
or down and to the
right.
In this case 1 down
and 1 right. Thus slope is -1/1 = -1
YOU TRY IT
Provide the equation of the
inequality.
YOU TRY IT: (Solution)
The inequality is solid and shaded below:
Y ≤ mx + b
1. Find the y-intercept
In this graph b = + 4.
2. Find another point
to get the slope.
A(0,4) B(1,0)
A(0,4)
B(1,0)
Use the equation of slope to find the slope:
−
−
=
−
−
=
−

=-4
The slope-intercept
form equation is:
A(0,4)
B(1,0)
y ≤ -4x + 4
Remember:
This means that if you start a 4 and move down
four and one over to the right. Solid line and
shaded down means we must use ≤.
When no work is required, you can use the
rise/run of a right triangle between the two
points:
Look at the triangle,
down 4 (-4) over to the
right 1 (+1)
slope = -4/+1 = -4
A(0,4)
B(1,0)
Remember:
You MUST KNOW BOTH procedures, the
slope formula and the triangle.
Given Two Points: We can also create an
inequality in the slope-intercept form from
any two points and the words: less than (<),
less than or equal to (≤), greater than(>),
greater than and equal to(≥) accordingly.
EX:
Write the slope-intercept form of the line
that is greater than or equal to and
inequality that passes through the points
(0, -0.5) and(2, -5.5)
Use the given points and equation of slope:
A(0,-0.5) B(2,-5.5)
−
−
=
−.−−.
−
=
−

= -

We now use the slope and a point to find the
y intercept (b).
y ≥ mx + b
-3 = Isolate b:

-3 + = b

b=- +

() + b
=-

Going back to the equation:
y = mx + b
we replace what we have found:

m = - and b = - ½
To get the final slope-intercept form of the
line passing through (3, -2) and(1, -3)
y≥

−

x–½
We now proceed to graph the equation:

y≥- x-

Y-intercept
y crossing
YOU TRY IT:
Write the equation of the inequality.
Use the given points and equation of slope:
B(1,-3)
A(3,-2)
−
−
=
−−−
−
=
−
−
=

We now use the slope and a point to find the
y intercept (b).
y < mx + b
-3 =
Isolate b:

() + b
-3 - = b
b=-

− =-

Going back to the equation:
y = mx + b
we replace what we have found:
m=

and b = -

To get the final slope-intercept form of the
line passing through (3, -2) and(1, -3)
y<

x -3.5
We now proceed to graph the equation:
y<

2
1
x-

Y-intercept
y crossing
Real-World:
A fish market charges \$9 per
pound for cod and \$12 per pound
per flounder. Let x = pounds of cod
and y = pounds of flounder. What
is the inequality that shows how
much of each type of fish the store
must sell per day to reach a daily
quota of at least \$120?
Real-World(SOLUTION):
A fish market charges \$9 per pound for cod
and \$12 per pound per flounder. Let x = pounds of
cod and y = pounds of flounder. What is the
inequality that shows how much of each type of
fish the store must sell per day to reach a daily
quota of at least \$120?
Cod
x
Flounder  y
At least \$120  9x + 12y ≥ 120
SOLUTION:
y≥-
9x + 12y ≥ 120 

x

+ 10
10
9
8
10
8
4
4
8
10
12
Any point in the line or in the shaded
region is a solution.
YOU TRY IT:
A music store sells used CDs
for \$5 and buys used CDs for \$1.50.
You go to the store with \$20 and
some CDs to sell. You want to have
at least \$10 left when you leave
the store. Write and graph an
inequality to show how many CDs
Real-World(SOLUTION):
A music store sells used CDs for \$5 and
buys used CDs for \$1.50. You go to the store
with \$20 and some CDs to sell. You want to
have at least \$10 left when you leave the
store. Write and graph an inequality to
show how many CDs you could buy and sell.
Bought CDs
Sold CDs
 -5x
 +1.5y
At least \$10 left  -5x + 1.5y ≥ -10
NOTE: -10 since you spent this money.
SOLUTION:
-5x + 1.5y ≥ -10 
y≥

x
.
– 6.6
10
8
6
4
2
1
2
3
4
Any point in the line or in the shaded
region is a solution.
VIDEOS:
Linear Inequalities
The text passage is a detailed explanation of graphing linear inequalities, including examples and real-world applications. It contains multiple questions and their corresponding solutions.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
6_5 Graphing Linear Inequalities

```6.5 Slope intercept form
for Inequalities:
Linear Inequality: is a linear equation with
an inequality sign (< , ≤, >, ≥)
Solution of an Inequality: is an ordered
pair (x, y) that makes the inequality true.
GOAL:
Whenever we are given a graph we must be able
to provide the equation of the function.
Slope-Intercept Form: The linear equation of a
nonvertical line with an inequality sign:
y (<, ≤, >, ≥) m x + b
Slope =
−
−
=
⍙
=
⍙
y-intercept
y crossing
http://mathgraph.idwvogt.com/examples.html
Whenever we are given a graph we must be able
to provide the equation of the function.
y<mx+b
dash line
Whenever we are given a graph we must be able
to provide the equation of the function.
y>mx+b
dash line
Whenever we are given a graph we must be able
to provide the equation of the function.
y≤mx+b
Solid line
Whenever we are given a graph we must be able
to provide the equation of the function.
y≥mx+b
Solid line
EX:
Provide the equation of the inequality.
Solution: Since line is dashed and shaded at
the bottom we use <. Also, the inequality
must be in
slope-intercept form:
Y < mx + b
1. Find the y-intercept
In this graph b = +1.
2. Find another point
to get the slope.
A(0,5)
A(0,1)
B(3,-2)
Use the equation of slope to find the slope:
−
−
=
−−
−
=
−

= -1
The slope-intercept
form inequality is:
A(0,1)
B(3,-2)
y < -1x + 1
Remember:
This means that if you start a 1 and move down
one and over to the right one, and continue this
pattern. We shade the bottom since it is <.
When work does not need to be shown:
(EOC Test)
look at the triangle
points.
Count the number
of square going up
or down and to the
right.
In this case 1 down
and 1 right. Thus slope is -1/1 = -1
YOU TRY IT
Provide the equation of the
inequality.
YOU TRY IT: (Solution)
The inequality is solid and shaded below:
Y ≤ mx + b
1. Find the y-intercept
In this graph b = + 4.
2. Find another point
to get the slope.
A(0,4) B(1,0)
A(0,4)
B(1,0)
Use the equation of slope to find the slope:
−
−
=
−
−
=
−

=-4
The slope-intercept
form equation is:
A(0,4)
B(1,0)
y ≤ -4x + 4
Remember:
This means that if you start a 4 and move down
four and one over to the right. Solid line and
shaded down means we must use ≤.
When no work is required, you can use the
rise/run of a right triangle between the two
points:
Look at the triangle,
down 4 (-4) over to the
right 1 (+1)
slope = -4/+1 = -4
A(0,4)
B(1,0)
Remember:
You MUST KNOW BOTH procedures, the
slope formula and the triangle.
Given Two Points: We can also create an
inequality in the slope-intercept form from
any two points and the words: less than (<),
less than or equal to (≤), greater than(>),
greater than and equal to(≥) accordingly.
EX:
Write the slope-intercept form of the line
that is greater than or equal to and
inequality that passes through the points
(0, -0.5) and(2, -5.5)
Use the given points and equation of slope:
A(0,-0.5) B(2,-5.5)
−
−
=
−.−−.
−
=
−

= -

We now use the slope and a point to find the
y intercept (b).
y ≥ mx + b
-3 = Isolate b:

-3 + = b

b=- +

() + b
=-

Going back to the equation:
y = mx + b
we replace what we have found:

m = - and b = - ½
To get the final slope-intercept form of the
line passing through (3, -2) and(1, -3)
y≥

−

x–½
We now proceed to graph the equation:

y≥- x-

Y-intercept
y crossing
YOU TRY IT:
Write the equation of the inequality.
Use the given points and equation of slope:
B(1,-3)
A(3,-2)
−
−
=
−−−
−
=
−
−
=

We now use the slope and a point to find the
y intercept (b).
y < mx + b
-3 =
Isolate b:

() + b
-3 - = b
b=-

− =-

Going back to the equation:
y = mx + b
we replace what we have found:
m=

and b = -

To get the final slope-intercept form of the
line passing through (3, -2) and(1, -3)
y<

x -3.5
We now proceed to graph the equation:
y<

2
1
x-

Y-intercept
y crossing
Real-World:
A fish market charges \$9 per
pound for cod and \$12 per pound
per flounder. Let x = pounds of cod
and y = pounds of flounder. What
is the inequality that shows how
much of each type of fish the store
must sell per day to reach a daily
quota of at least \$120?
Real-World(SOLUTION):
A fish market charges \$9 per pound for cod
and \$12 per pound per flounder. Let x = pounds of
cod and y = pounds of flounder. What is the
inequality that shows how much of each type of
fish the store must sell per day to reach a daily
quota of at least \$120?
Cod
x
Flounder  y
At least \$120  9x + 12y ≥ 120
SOLUTION:
y≥-
9x + 12y ≥ 120 

x

+ 10
10
9
8
10
8
4
4
8
10
12
Any point in the line or in the shaded
region is a solution.
YOU TRY IT:
A music store sells used CDs
for \$5 and buys used CDs for \$1.50.
You go to the store with \$20 and
some CDs to sell. You want to have
at least \$10 left when you leave
the store. Write and graph an
inequality to show how many CDs
Real-World(SOLUTION):
A music store sells used CDs for \$5 and
buys used CDs for \$1.50. You go to the store
with \$20 and some CDs to sell. You want to
have at least \$10 left when you leave the
store. Write and graph an inequality to
show how many CDs you could buy and sell.
Bought CDs
Sold CDs
 -5x
 +1.5y
At least \$10 left  -5x + 1.5y ≥ -10
NOTE: -10 since you spent this money.
SOLUTION:
-5x + 1.5y ≥ -10 
y≥

x
.
– 6.6
10
8
6
4
2
1
2
3
4
Any point in the line or in the shaded
region is a solution.
VIDEOS:
Linear Inequalities
user
The text passage is a detailed explanation of graphing linear inequalities, including examples and real-world applications. It contains multiple questions and their corresponding solutions.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS