6.5 Slope intercept form for Inequalities:
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Transcript of 6.5 Slope intercept form for Inequalities:
6.5 Slope intercept formfor 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:
Slope = =
y (<, ≤, >, ≥) m x + b
y-intercepty 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 < m x + b dash line shade left or down
Whenever we are given a graph we must be able to provide the equation of the function.
y > m x + b dash line shade right or up
Whenever we are given a graph we must be able to provide the equation of the function.
y ≤ m x + b Solid line shade left or down
Whenever we are given a graph we must be able to provide the equation of the function.
y ≥ m x + b Solid line shade Right or up
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(0,1)
B(3,-2)
A(0,5) B(3,-2)
Use the equation of slope to find the slope:
= = = -1
The slope-intercept form inequality is:
y < -1x + 1Remember: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 <.
A(0,1)
B(3,-2)
When work does not need to be shown: (EOC Test) look at the triangle made by the two 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:
y ≤ -4x + 4Remember: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 ≤.
A(0,4)
B(1,0)
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:
= = = -
We now use the slope and a point to find the y intercept (b).
A(0,-0.5) B(2,-5.5)
y ≥ mx + b -3 = - + b
Isolate b: -3 + = b
b = - = -
Going back to the equation:
y = mx + b
m = - and b = - ½
To get the final slope-intercept form of the line passing through (3, -2) and(1, -3)
we replace what we have found:
y ≥ x – ½
We now proceed to graph the equation:
y ≥ - x - 𝑹𝒊𝒔𝒆𝑹𝒖𝒏
Y-intercepty crossing
YOU TRY IT: Write the equation of the inequality.
Use the given points and equation of slope:
= = =
We now use the slope and a point to find the y intercept (b).
A(3,-2) B(1,-3)
y < mx + b -3 = + b
Isolate b: -3 - = b
b = - = -
Going back to the equation:
y = mx + b
m = and b = -
To get the final slope-intercept form of the line passing through (3, -2) and(1, -3)
we replace what we have found:
y < x -3.5
We now proceed to graph the equation:
y < x - 𝑹𝒊𝒔𝒆𝑹𝒖𝒏
Y-intercepty crossing
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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):
Cod x Flounder y
At least $120 9x + 12y ≥ 120
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?
SOLUTION: 9x + 12y ≥ 120 y ≥ - x + 10
4 8
4
8
10
8910
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 you could buy and sell.
Real-World(SOLUTION):
Bought CDs -5x Sold CDs +1.5y
At least $10 left -5x + 1.5y ≥ -10
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.
NOTE: -10 since you spent this money.
SOLUTION: -5x + 1.5y ≥ -10 y ≥ x – 6.6
1 2
2
4
6
3 4
Any point in the line or in the shaded region is a solution.
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VIDEOS: Linear Inequalities
https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/graphing-linear-inequalities/v/solving-and-graphing-linear-inequalities-in-two-variables-1
https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/graphing-linear-inequalities/v/graphing-inequalities
CLASSWORK:
Page 393-395
Problems: As many as needed to master the
concept.