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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 shade left or down Whenever we are given a graph we must be able to provide the equation of the function. y>mx+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≤mx+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≥mx+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(0,5) 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 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: 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 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: -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 https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/graphing-linearinequalities/v/graphing-inequalities https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/graphing-linearinequalities/v/solving-and-graphing-linearinequalities-in-two-variables-1 CLASSWORK: Page 393-395 Problems: As many as needed to master the concept.