Download presentation

Presentation is loading. Please wait.

Published byElinor Lawrence Modified over 6 years ago

1
Linear Inequalities Foundation Part I

2
An INEQUALITY shows a relationship between two variables, usually x & y Examples –y > 2x + 1 –y < x – 3 –3x 2 + 4y ≥ 12 What is an inequality?

3
Objective of these next few slides Read a graph and write down the inequalities that contain a region Draw inequalities and indicate the region they describe You need to know how to plot straight line graphs (yesterday)

4
For example x y x > 2 X=2 When dealing with ONE inequality, we SHADE IN the REQUIRED REGION

5
For example x y x < -2 X=-2

6
For example x y y < -1 y=-1

7
For example x y y < 2x +1 y= 2x+1 Which side is shaded? Pick a point NOT on line (1,2) Is 2 < 2 x 1 + 1 ? YES (1,2) lies in the required region

8
For example x y y > 3x - 2 y= 3x-2 Which side is shaded? Pick a point NOT on line (2,1) Is 1 > 3 x 2 - 2 ? NO (2,1) does NOT lie in the required region

9
How to draw graph of equation x y y = 3x + 2 Shade IN the Region for y > 3x + 2 (2,1) Is 1 > 3 x 2 + 2 ? NO (2,1) does NOT lie in the required region

10
How to draw graph of equation x y 4y + 3x = 12 Shade IN the Region for 4y + 3x > 12 (3,2) Is 4 x 2 + 3 x 3 > 12 ? YES (3,2) DOES lie in the required region

11
Regions enclosed by inequalities

12
y = 3 x = 4 x + y = 4 y < 3 x < 4 x + y > 4 (3,2) 2 < 3 ? 3 + 2 > 4 ? 3 < 4 ?

13
Part II Solving Linear and Quadratic Inequalities

14
Linear Inequalities These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality. Consider the numbers 1 and 2 : Examples of linear inequalities: 1.2. Dividing or multiplying by 1 gives 1 and 2 BUT 1 is greater than 2 So, We know ( 1 is less than 2 ) 11 22

15
Linear Inequalities These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality. Examples of linear inequalities: 1.2.

16
Exercises Find the range of values of x satisfying the following linear inequalities: 1. 2. Solution: Solution: Either OrDivide by -4: so,

17
Quadratic Inequalities Solution: e.g.1 Find the range of values of x that satisfy Rearrange to get zero on one side: or is less than 0 below the x -axis The corresponding x values are between -3 and 1 Let and solve Method: ALWAYS use a sketch

18
Solution: e.g.2 Find the values of x that satisfy or There are 2 sets of values of x Find the zeros of where is greater than or equal to 0 above the x -axis or These represent 2 separate intervals and CANNOT be combined

19
Solution: e.g.3 Find the values of x that satisfy Find the zeros of where is greater than 0 above the x -axis This quadratic has a common factor, x or Be careful sketching this quadratic as the coefficient of is negative. The quadratic is “upside down”.

20
Linear inequalities Solve as for linear equations BUT Keep the inequality sign throughout the working If multiplying or dividing by a negative number, reverse the inequality Quadratic ( or other ) Inequalities rearrange to get zero on one side, find the zeros and sketch the function Use the sketch to find the x -values satisfying the inequality Don’t attempt to combine inequalities that describe 2 or more separate intervals SUMMARY

21
Exercise or There are 2 sets of values of x which cannot be combined is greater than or equal to 0 above the x -axis or 1. Find the values of x that satisfy where Solution:

22
Now do Exercise 4A page 126

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google