Linear Inequalities | Mathematics Form 2

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Linear Inequalities - Mathematics Form 2













Objectives

    By the end of the lesson you should be able to:

    • Solve the linear inequalities in two unknown geographically
    • Form simple linear inequalities from inequality graphs.

Below are inequality symbols


use is illustrated below.

and when changed


LESSON INTRODUCTION
Graphical representation of linear inequalities with one unknown.

An inequality represents a region with one or more boundaries.
Consider the inequality y < 4.
To get the boundary line, the inequality symbol (<) is replaced with the equals (=) symbol. Therefore the boundary line for y<4 is y=4. A boundary line can be drawn as a broken or a continuous line.
For an equality with the symbol > or <, the boundary line us drawn using a broken line. This means that points on the line are not included in the required region.

Example 1
Represent y < 4 graphically
Solution
Draw the boundary line y = 4 (use a broken line)

Diagram 4

To represent the inequality, choose two points (one above and one below the boundary line) and substitute each point in the inequality. The point that satisfies the inequality is in the wanted region and the point that doesnt satisfy the inequality is in the unwanted region. The unwanted region is always shaded.

Consider points (1,1) and (2,5). Point (1,1) is below the boundary line y=4 and the y-value (1) is less than (4). Therefore point (1,1( is in the wanted region since it satisfies the inequality y<4.

Similarly for point (2,5) the y value is 5 which is greater than 4. This shows that point (2,5) is in the unwanted region since it does not satisfy the inequality.

This means the region above the boundary line is the unwanted region is shaded as shown below.

  

Diagram 5
Example 2
Represent 2

Solution
The boundary line is y=2.
For an inequality with the symbols 3, the boundary lines ate drawn using a continuous line. This means that points on the boundary line are included in the wanted region. Draw the boundary line y=2 (use a continuous line)


Note;

    1. The shaded region is the unwanted region
    2. The points on the boundary satisfy the inequality 1.
    3. The points on the line satisfy the inequality 2.

Example 3
Represent 2graphically.

Solution
The boundary line is x=2 and the wanted region if the one where an x value is equal to ot greater than 2.


Example 4
Represent x<-3 graphically.

Solution
The boundary line is x= -3 and the wanted region is the one where an x value is less than 3.

LESSON DEVELOPMENT


1.   Graphical representation of Inequalities involving two unknowns.

   Example
Draw a graph of the inequality 3x + 4y,12.

Solution
The boundary line is 3x + 4y = 12. Make a table of values of x and y for 3x + 4y = 12 on the Cartesian plane. (Draw the boundary line using a broken line)

Plot this on the Cartesian plane.

     X    0    4
     Y    3  0


Use bright colour to shade and draw line

              
Substituting (1,1) in the equation 3(1) + 4(1) < 12 we het 7 which is less than 12 which is below the boundary line. Point (1,1) which is below the boundary line wanted region. Therefore the region above the line is shaded.


Solving Simultaneous Linear Inequalities Graphically

To solve simultaneous inequalities draw the inequalities in one graph. The solution are given by points in the wanted region.

Example
Draw graph of:  
y4x +3,   x + y <4 and y30 and show the region that satisfy the inequalities.

Solution
The boundary lines are y = x + 3, x + y = 4 and y = 0.
The tables of values are as shown below.
(i)  y = x + 3
x    -3   0    x    0    4
y    0    3    y    4    0

The following is the required graph.
(Use bright colours to draw graph)

R is out required region

Diagram 10 (Note that the wanted region indicated by letter R.)

2.   INEQUALITIES FROM INEQUALITIES GRAPHS

In this lesson, we shall define a region which is already graphed by determining the inequalities that satisfies the region.
Example
State the inequalities that represent the unshaded region R in each of the following areas.

Case 1 (+)


The gradient of line 1 is 0. The line cuts the y-axis at 2. Therefore using y = mx + c, the equation of the line of all the points in the line y = 2 is always 2.

Similarly, line 2 cuts x-axis at 3 and the x-coordinates of all the points on the line is always 3. Therefore the required line is x = 3.
Considering line 1 (y=2) choose a point (0,0) in the wanted region R. The value of y is 0. Therefore (0,0) is in the wanted region which has y values that are less than 2.

Note that, the boundary line is broken. Therefore, the required inequalities is y<2. Considering line 2 (x=3) point x value (1) is less than 3.

Therefore point (1,2) is in the region with x values that are less than 3. Note that the line is a continuous line. Therefore, the inequality is x53

Step 1

To find the equations for the boundary lines A,B and C.

Line A
Identify two points on the line
(3,0) and (0,2)
 Find the gradient (M) of the line
67876
2

Let P(x,y) be a general point on the line A.  Using the point (3,0)
3

2    3

Alternatively
Use the equation of strait line y = mx+C
Where the line A passes through (0,2) on the y-axis.

The y-intercept (c) is 2 and the gradient (M) is 3, using the point P(x,y) and substituting  in y = mx + c,

The equation becomes
      3

For line B
The line passes through points (0,2) and (1,3), Point (0,2) is on the y axis.
Using y = mx + C (equation of a general straight line) C is 2.
3
3

The equation is 3 
      = y = 3+ 2

Line C
The line passes through points e.g.
(-1,0), (-1, 1),  (-1, 2) etc

The x- value in all the points on the line is -1 so the equation is X = -1

Step 2

To get the inequalities

Line A  33 Using (0, 0) which is in the wanted region R. Substitute it in the equation 3 using point (0,0) and after substituting 3

Therefore 0 < 2,

The inequality is 3

Line B, y = x + 2(-0.5, 2)
Using the point which is in the required region R

   Y = 2,   and when substituted in the equation,   y = -0.5 + 2 = 1.5

and 2 > 1.5
The inequality is
    
   Y > x + 2

For line C(x = -1) using points (-0.5, 2) and (0,2), the x-value is 2 which is greater than 1. Therefore the inequality is x>-1.

In summary region R is defined by the inequalities

i)    y =  -?x + 2

ii)   y>x + 2

iii)   x > -1


LESSON DEVELOPMENT

  Graphical representation of Inequalities involving two unknowns.

  
    Example


Draw a graph of the inequality 3x + 4y,12.

Solution
The boundary line is 3x + 4y = 12. Make a table of values of x and y for 3x + 4y = 12 on the Cartesian plane. (Draw the boundary line using a broken line)

Plot this on the Cartesian plane.

     X    0    4
     Y    3  0


Use bright colour to shade and draw line

              
Substituting (1,1) in the equation 3(1) + 4(1) < 12 we het 7 which is less than 12 which is below the boundary line. Point (1,1) which is below the boundary line wanted region. Therefore the region above the line is shaded.


Solving Simultaneous Linear Inequalities Graphically

To solve simultaneous inequalities draw the inequalities in one graph. The solution are given by points in the wanted region.

Example
Draw graph of:  
y1x +3,   x + y <4 and y20 and show the region that satisfy the inequalities.

Solution
The boundary lines are y = x + 3, x + y = 4 and y = 0.
The tables of values are as shown below.
(i)  y = x + 3
x    -3   0    x    0    4
y    0    3    y    4    0

The following is the required graph.
(Use bright colours to draw graph)

R is out required region

Diagram 10 (Note that the wanted region indicated by letter R.)

INEQUALITIES FROM INEQUALITIES GRAPHS

In this lesson, we shall define a region which is already graphed by determining the inequalities that satisfies the region.
Example
State the inequalities that represent the unshaded region R in each of the following areas.

Case 1 (+)

The gradient of line 1 is 0. The line cuts the y-axis at 2. Therefore using y = mx + c, the equation of the line of all the points in the line y = 2 is always 2.

Similarly, line 2 cuts x-axis at 3 and the x-coordinates of all the points on the line is always 3.

Therefore the required line is x = 3.
Considering line 1 (y=2) choose a point (0,0) in the wanted region R. The value of y is 0. Therefore (0,0) is in the wanted region which has y values that are less than 2.

Note that, the boundary line is broken. Therefore, the required inequalities is y<2. Considering line 2 (x=3) point x value (1) is less than 3.

Therefore point (1,2) is in the region with x values that are less than 3. Note that the line is a continuous line. Therefore, the inequality is x13

Step 1

To find the equations for the boundary lines A,B and C.

Line A
Identify two points on the line
(3,0) and (0,2)
 Find the gradient (M) of the line
2
3

Let P(x,y) be a general point on the line A.  Using the point (3,0)
4

5    6

Alternatively
Use the equation of strait line y = mx+C
Where the line A passes through (0,2) on the y-axis.

The y-intercept (c) is 2 and the gradient (M) is 8, using the point P(x,y) and substituting  in y = mx + c,

The equation becomes
      7

For line B
The line passes through points (0,2) and (1,3), Point (0,2) is on the y axis.
Using y = mx + C (equation of a general straight line) C is 2.
9
10

The equation is 11 
      = y = 12+ 2

Line C
The line passes through points e.g.
(-1,0), (-1, 1),  (-1, 2) etc

The x- value in all the points on the line is -1 so the equation is X = -1

Step 2

To get the inequalities

Line A  13 Using (0, 0) which is in the wanted region R. Substitute it in the equation 14 using point (0,0) and after substituting 15

Therefore 0 < 2,

The inequality is 16

Line B, y = x + 2(-0.5, 2)
Using the point which is in the required region R

   Y = 2,   and when substituted in the equation,   y = -0.5 + 2 = 1.5

and 2 > 1.5
The inequality is
    
   Y > x + 2

For line C(x = -1) using points (-0.5, 2) and (0,2), the x-value is 2 which is greater than 1. Therefore the inequality is x>-1.

In summary region R is defined by the inequalities

i)    y =  -?x + 2

ii)   y>x + 2

iii)   x > -1


Graphs of Inequalities in One Unknown

Graphical Representation of Linear Inequalities

Graphical Representation of Inequalities in Two Unknowns

Inequalities from Inequality Graphs

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