Graph linear inequalities in two variables Section 6.7 #44 There is nothing strange in the circle being the origin of any and every marvel. Aristotle

Concept Up until this point we’ve discussed inequalities that involve only one dimension or one variable Up until this point we’ve discussed inequalities that involve only one dimension or one variable Today we’re going to take our understanding of inequalities and apply it to two dimensions (variables) Today we’re going to take our understanding of inequalities and apply it to two dimensions (variables) First we will do a short review of lines and linear terms First we will do a short review of lines and linear terms

Slope Slope is Slope is A. An index of the angle of a line B. A ratio of how much a line increases versus how much to moves right of left C. A ratio of run to rise D. An index of movement in the x direction

Slope What is the slope of the line that goes through the points (1,2) & (5,4) What is the slope of the line that goes through the points (1,2) & (5,4)

Slope What is the slope of the line that goes through the points (-4,-2) & (7,-8) What is the slope of the line that goes through the points (-4,-2) & (7,-8)

Slope What is the slope of the line that goes through the points & What is the slope of the line that goes through the points &

Slope What is the slope of the line that goes through the points (5,2) & (5,4) What is the slope of the line that goes through the points (5,2) & (5,4)

Slope What is the equation of the line that goes through the points (1,3) & (3,7) What is the equation of the line that goes through the points (1,3) & (3,7)

Slope What is the equation of the line that goes through the points (-3,4) & (-5,-12) What is the equation of the line that goes through the points (-3,4) & (-5,-12)

Slope What is the equation of the vertical line that goes through the point (3,-5) What is the equation of the vertical line that goes through the point (3,-5)

Slope The equation of a line is y=3x-9. The slope of the line is increased by 2. What happens to the line? The equation of a line is y=3x-9. The slope of the line is increased by 2. What happens to the line? A. The line has the same y-intercept, but now slopes downward B. The line has the same y-intercept, but is now steeper C. The line has a different y-intercept, but now slopes downward D. The line has a different y-intercept, but is now steeper

Slope Assuming that the line starts at x=0, which line will reach y=50 first? Assuming that the line starts at x=0, which line will reach y=50 first?

The big idea When we look at a line, we’re seeing the collection of points that are solutions to a linear equality When we look at a line, we’re seeing the collection of points that are solutions to a linear equality When looking at a linear inequality, instead of looking at a set of points, we are seeing a defined space that indicates the infinite collection of points that satisfy the criteria When looking at a linear inequality, instead of looking at a set of points, we are seeing a defined space that indicates the infinite collection of points that satisfy the criteria For example For example Y X This means that any point that falls in the shaded area is a viable solution to the inequality

Testing a point We can see this by testing out a point in the shaded area We can see this by testing out a point in the shaded area For example For example Y X (-6,3) It’s imperative that we remember that the solution to these inequalities is an area as opposed to a line

Process out of examples Our process for creating these graphs is not difficult, but rather just an extension of our previous knowledge of graphing Our process for creating these graphs is not difficult, but rather just an extension of our previous knowledge of graphing Y X Graph the line via linear graphing methods Draw a dashed line for >,< otherwise a solid line Shade the appropriate area Above for greater than Below for less than

Example Let’s do an example Let’s do an example Y X

Example How would we graph this one? How would we graph this one? Y X

Example We would operate horizontal and vertical inequalities the same as any other inequality We would operate horizontal and vertical inequalities the same as any other inequality Y X

Examples Y X

Example And another one And another one Y X

Example Y X

Practical Example A party shop makes giftbags for birthday parties. They charge $4 per glowstick and $10 per T-shirt. Let x represent the number of glowsticks and y the number of T-shirts. The goal is to earn at least $500 from the sale of the bags Write an inequality that describes the goal in terms of x & y Graph the inequality Give three possible combinations of pairs that will allow the shop to meet it’s goal Y X

Most Important Points What’s the most important thing that we can learn from today? What’s the most important thing that we can learn from today? The solution to an inequality in two-dimensions is an area, as opposed to a line The solution to an inequality in two-dimensions is an area, as opposed to a line We can graph the solutions to an equation by following our normal processes for graphing lines and then shading the appropriate area We can graph the solutions to an equation by following our normal processes for graphing lines and then shading the appropriate area

Homework 6.7 you will have two days 1, 2-32, 47-50, 53-57