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. ymx+b

dash line shade right or up Whenever we are given a graph we must be able to provide the equation of the function. ymx+b Solid line shade left or down

Whenever we are given a graph we must be able to provide the equation of the function. ymx+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 A(0,1) 1. Find the y-intercept In this graph b = +1. 2. Find another point to get the slope. A(0,5) B(3,-2)

B(3,-2) Use the equation of slope to find the slope: = = = -1 A(0,1) The slope-intercept form inequality is: 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 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

A(0,4) The slope-intercept form equation is: 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 = - + b

Isolate 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 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: A(3,-2) = = = B(1,-3) We now use the slope and a point to find the

y intercept (b). y < mx + b -3 = + b Isolate 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< x

2 1 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 - x + 10 9x + 12y 120 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

you could buy and sell. 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: y x 6.6 -5x + 1.5y -10

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 https:// www.khanacademy.org/math/algebra/linear-equa tions-and-inequalitie/graphing-linear-inequalities/

v/graphing-inequalities https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/graphing-linear-inequ alities/v/solving-and-graphing-linear-inequalities-i n-two-variables-1 CLASSWORK: Page 393-395 Problems: As many as needed to master the concept.