• Share
  • Email
  • Embed
  • Like
  • Save
  • Private Content
New approaches in linear inequalities
 

New approaches in linear inequalities

on

  • 657 views

 

Statistics

Views

Total Views
657
Views on SlideShare
657
Embed Views
0

Actions

Likes
0
Downloads
7
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Microsoft Word

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

    New approaches in linear inequalities New approaches in linear inequalities Document Transcript

    • TARUNGEHLOTS an Inquiry Based Approach to Solving a System of Linear Inequalities This lesson is designed to help the students discover and work with basic inequalities. The lesson is designed to guide the students through graphing inequalities on a number line and using X and Y-intercepts to graph linear equations on the coordinate grid. The lesson will then have the students graph linear inequalities and correctly shade the solution area on the coordinate grid. This will then lead into having the student use X and Y-intercepts to graph a system of linear inequalities on the coordinate grid and correctly shade the solution area. Finally, the lesson will conclude with the students having to locate a treasure chest by using a system of linear inequalities.Plotting inequalities on a number lineName________________________________________Goal: This lesson is designed to help you work with and understand basic inequalities. 1. Given the statement: The temperature will be higher than 70 degrees today. A) Give three possible temperatures that satisfy the statement. B) We could represent this by using an inequality. (x > 70) C) We could also represent this with a number line graph (Notice the open circle above 70. This indicates that 70 is not included in the values represented on the graph since we only wanted temperatures larger than 70) ○ <-----|-----|-----|-----|-----|-----|-----|-----> 40 50 60 70 80 90 100 2. Given the statement: To ride the Little Dipper you must be under 48 inches tall. A) Give three possible heights that satisfy this statement B) Write an inequality to represent the statement. C) Notice that the number line graph is going the opposite direction since the inequality is different. ○ <-----|-----|-----|-----|-----|-----|-----|-----> 44 46 48 50 52 54 56 -1-
    • 3. Given the statement: The temperature will be 70 degrees or warmer today. A) Give three possible temperatures that satisfy the statement. B) We could represent this by using an inequality. (x > 70) C) What is different between the graph of this statement and the statement in #1?(look closely at the circles) Why do you think this difference is necessary? ● <-----|-----|-----|-----|-----|-----|-----|-----> 40 50 60 70 80 90 100 4. Given the statement: To ride the Little Dipper you must be 48 inches tall or shorter. A) Give three possible heights that satisfy this statement B) Write an inequality to represent the statement. C) If you were drawing the number line graph, would you use an open or closed circle? Why? D) Would your line go to the right or left? Why? E) Draw the number line graph. <-----|-----|-----|-----|-----|-----|-----|-----> 44 46 48 50 52 54 56 5. If x > 5, what does this mean? What x values make the statement true? 6. Why is 5 not included in your list? 7. If x < 2, what does this mean? What x values make the statement true? 8. Why is 2 included in this list? 9. Complete the following tableSymbol Meaning Open or closed Circle?><>> 10. To graph x > 4 on a number line, we would need to show all the numbers that are greater than but not equal to 4. Draw a number line graph for x > 4 -2-
    • <-----|-----|-----|-----|-----|-----|-----|-----> 0 1 2 3 4 5 611. Graph x < 2 on a number line. <-----|-----|-----|-----|-----|-----|-----|-----> 0 1 2 3 4 5 612. Plot the following ordered pairs on the same coordinate grid and then connect the points with a straight line. (2,6), (-3,6), (-7,6), (3,6)What do you notice about your line? __________________________________What do you notice about your ordered pairs? ___________________________13. Graph y = 6 on the coordinate grid. -3-
    • 14. Compare your graphs from #5 and #6. What do you notice about them? _________________________________________________________What type of line do you get when all your y values are the same? ____________15. Plot the following ordered pairs on the same coordinate grid and then connect the points with a straight line. (3,7), (3,-4), (3, -5), (3,4) -4-
    • What do you notice about your line? ___________________________________What do you notice about your ordered pairs? ____________________________16. Graph x = 3 on the coordinate grid.17. Compare your graphs from #7 and #8. What do you notice about them? _________________________________________________________What type of line do you get when all your x values are the same? ____________ -5-
    • Look at the following graphs and answer the questions.A) List 3 ordered pairs for each graphB) Write an equation of the line represented by the graph18. 19. -6-
    • Day 1 Recap Worksheet 1. Write an inequality for the statement The temperature will be cooler than 50 degrees tonight. 2. Draw a number line graph to represent the statement in question 1. <-----|-----|-----|-----|-----|-----|-----|-----> 3. Draw a number line graph to represent the inequality x > -4 <-----|-----|-----|-----|-----|-----|-----|-----> 4. On a coordinate grid, graph the line y = -2 5. Name three ordered pairs on your line. 6. The graph of x = -5 will be a horizontal/vertical (choose one) line. -7-
    • Graphing the standard form of linear equations on the coordinate grid.Name_______________________________________Goal: This lesson is designed to let you discover how to use X and Y-intercepts to graphlinear equalities on the coordinate grid.Intercepts are where one or more objects cross each other. When we talk about the X-intercept we are concerned with the point where the graph crosses the X-axis. Likewise,when we talk about the Y-intercepts of a graph, we are concerned with the point wherethe graph crosses the Y-axis.The graph in the above picture crosses the X-axis at point (5,0) therefore we say the X-intercept is 5. It crosses the Y-axis at point (0,-4) and we say that the Y-intercept is -4.Using your knowledge of plotting points on the coordinate grid, plot (0,3) and (-2,0).Draw a line through these points.What conclusions can you draw about (0,3) and (-2,0)? -8-
    • Where is the graph crossing the X-axis?________ What is the X-intercept?_______Where is the graph crossing the Y-axis?________ What is the Y-intercept?_______Using the above picture, what do you notice about the coordinates where the graphcrosses the X and Y-axis?You should have noticed that at the X-intercept the Y value is always zero and at the Y-intercept the X value is always zero. Now lets use this information to quickly constructgraphs of linear equations that are in the written in the standard form.Take a look at 4x + 6y = 12. In order to find the X-intercept we know that the Y valuemust equal zero. Therefore, if we substitute zero in for y we now have: 4x + 6(0) = 12.This then gives us 4x = 12. Solving for x we get x = 3. Thats our X-intercept!Lets find the Y-intercept.Remember that when we are looking for an intercept, one value must always equal zero.Since we are looking for the Y-intercept our X value must equal zero.Therefore, 4(0) + 6y = 12. This gives us 6y = 12, and y = 2. Thats our Y-intercept.Now lets graph the line. Our X-intercept is 3 so we plot a point at 3 on the positive sideof the X-axis and our Y-intercept is 2 so we plot a point at 2 on the positive side of the Y-axis. Now connect the dots (remember to extend your line through these points). -9-
    • Your Turn!Graph 10x + 5y = 20What is the X-intercept? (hint: where doesit cross the X-axis?) ______What is the Y-intercept?______Did you plot the points correctly andremember to extend your line through thepoints? - 10 -
    • Activity worksheet.Graph the following equalities using the X and Y-intercepts.3x + 3y = 9What is your X-intercept?_______What is your Y-Intercept?_______7x – 2y = 14What is your X-intercept?_______What is your Y-Intercept?_______-4x – 8y = 24What is your X-intercept?_______What is your Y-Intercept?_______ - 11 -
    • Graphing the standard form of linear inequalities on the coordinate grid.Name______________________________________Goal: This lesson is designed to let you discover how to use X and Y-intercepts to graphlinear inequalities on the coordinate grid and correctly shade the solution area.When we are graphing linear inequalities that are written in the standard form, we followthe same format of using the X and Y-intercepts. There are several very importantdifferences with the actual graph though. Lets take a look!3x + 2y > 6On the right is the graph of the above linearinequality. What do you notice about thegraph?4x – 12y ≥ 24On the right is the graph of the above linearinequality. What do you notice about thegraph?What is different between this graph and thefirst graph?Just like when we graphed numbers on a number line and had closed and open circlesdepending on the inequality used, the same rule applies here. Instead of circles though,now we have lines. When we are using the < and > inequalities, our graph is a dottedline. When we are using ≤ and ≥ inequalities, our graph is a solid line. The solid lineindicates that we include the points on the line. - 12 -
    • Now you try it!-2x + 4y < 12What is your X-intercept?________What is your Y-intercept?________Did you remember to use the correct linestyle?Lets go take a look at the following inequality.3x + 2y ≥ 6Notice that graph cuts the coordinate planeinto two pieces. One above the graph andone below the graph. Remember that whenwe are dealing with inequalities, only certainvalues actually "work" in the inequality.These values that work, make the inequalitytrue and those values that dont work makethe inequality false. In order to find outwhich work and which dont, we pick testpoints. Test points are any points that lieabove or below the graph. Lets pick thepoint (0,0) and find out if it works or not in the inequality.If we substitute (0,0) into the inequality we get 3(0) + 2(0) ≥ 6. Solving we get 0 + 0 ≥ 6,which gives us 0 ≥ 6. Obviously this is false and therefore (0,0) does not work in theinequality. Notice that (0,0) is located below our graph.Now lets try a point above our graph. How about (4,4) (it doesnt matter what point justso long as it is above the graph). If we substitute (4,4) into our inequality we get 3(4) +2(4) ≥ 6. Solving we get 12 + 8 ≥ 6 and this gives us 20 ≥ 6. Obviously true.What do you think will happen if you pick any point below the graph?What do you think will happen if you pick any point above the graph?What do you think will happen if you pick any point on the graph? - 13 -
    • The correct representation of the previous inequality is graphed below.Wow, now some of the coordinate plane isshaded.Why do you think just the area above the graphis shaded?If you said because that is where all the truevalue are, youre correct. When we aregraphing inequalities, you must shade the areaof the coordinate plane that contains the orderedpairs that work in the inequality.Lets try one more together. How about -2x + 6y ≤ 18.Based on the inequality, what type of line will we have (solid or dotted)?___________Start with the X and Y-intercepts.X-intercept is _______________________Y-intercept is _______________________Now lets plot those points and connect the dots with a _____________ line (solid ordotted).Pick a test point. I always like to use (0,0) whenit is available.If we substitute (0,0) into our inequality, do weend up with a true or false answer?What side of the coordinate plane should weshade, above or below?Did you shade it? - 14 -
    • Now its your turn all by yourself.6x – 3y > 244x + 2y ≥ -8 - 15 -
    • 5x -15y < 30 - 16 -
    • Graphing and solving a system of linear inequalities on the coordinate grid.Name__________________________________________Goal: This lesson is designed to let you discover how to use X and Y-intercepts to grapha system of linear inequalities on the coordinate grid and correctly shade the solutionarea.In the previous lesson you learned how to graph and shade a linear inequality. Now youwill learn how to graph and shade a system of linear inequalities. A system of linearinequalities is more than one linear inequality that will have common solutions. Throughgraphing and shading you will be able to discover the area that represents the commonsolutions. Lets get started!As mentioned before, a system is more than one inequality. Lets start with two linearinequalities.3x + 9y > 274x -2y ≤ 12Approach these inequalities one at a time.Find the X and Y-intercepts of the firstproblem.X-intercept_______Y-intercept_______Plot those points and draw your graph. Didyou remember to use the correct type of line?Now pick a test point and shade the correctportion of the coordinate plane with the blue colored pencil.Time to graph the next inequality. Start with the intercepts.X-intercept_______________Y-intercept_______________Plot the points, draw your graph with the correct type of line and pick a test point that iseither above or below the linear inequality you just graphed, not the one you graphedearlier. Shade the correct side with the yellow colored pencil.Are there any areas that contain both blue and yellow? - 17 -
    • In order to solve a system, the solution must satisfy each equation in the system. In ourcase, the inequalities must both be true. Lets pick a test point (how about (0,0)) andsubstitute it into both inequalities. The first inequality will give us 0 > 27 and the secondwill give us 0 ≤ 12. The first is false and the second is true.What do you notice about the shading at (0,0)?Now lets try (1,8). The first inequality gives us 75 > 27 and the second inequality givesus -12 ≤ 12. Both of these are true!What do you notice about the shading at (1,8)?What conclusions can you draw about shading and solutions to the system?Do you think anything changes if there are three, four, five or more inequalities in oursystem? If you answered no, youre right. Besides a little more work, nothing changes.Lets try a system with 4 inequalities.7x + 5y < 358x – 7y ≤ 562x + 3y > -125x – 3y ≥ -15 - 18 -
    • An archeologist has discovered a series of coordinates that mayrepresent the location of buried treasure. After deciphering thecoded messages, it seems that the only true coordinates of thetreasures are those coordinates that lie within the solution of thesystems of inequalities.Potential Treasures may be located at the coordinates designated on the mapbelow. Record the possible locations of the treasures on the lines provided.1. ______________ 2. ______________ 3. ________________ 4._____________5. ______________ 6. ______________ 7. ________________ 8. _____________ - 19 -
    • The solution to the following system of inequalities is the location of thetreasure. You will need to graph the inequalities on the provided grid. Thetreasure will be located at the point that lies within the solution of the systemof inequalities.2x + 3y < 18-4x – 4y < 8-3x + 4y < 124x – 5y < 20 - 20 -