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

12

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.

8

10

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.