In this post we will try to understand how to solve inequality using graphical representation. In order to do that, first you need to have basic understanding of how to plot the equation in the graph.
How to plot equation in graph
Let ax + by=c is the form of linear equation ( as highest power of x is 1)
Remember that the graph of linear equation is straight line
Step 01:
Put y = 0 and then find the value of x
ax + b(0) =c
ax = c
x = c/a
Plot this point in two dimensional plane
Step 02:
Put x = 0 and then find the value of y
a(0) + by = c
by = c
y = c/b
Plot this point in two dimensional plane
Step 03
At this point you have got two points in the graph
x = c/a and y = c/b
Join this two point with the help of straight line, you have now got the graphical representation of the equation ax + by = c
This is how we can plot the graphical representation of linear equation. Let us now see some more examples for our understanding
(a) Plot 2x + 4y = 8
Step 01
Put y = 0
we get x= 4
Step 02
Put x = 0
we get y =2
Step 03
Plot these two coordinates (4, 0) and (0, 2) in the graph chart and join them using straight line
(b) Plot 2x – y =4
Step 01
Put y = 0
we get x = 2
Step 02
Put x =0
we get y= -4
Step 03
Plot (2,0) and (0, -4) in the graph and join the points with the help of straight line
I hope by analyzing the above examples you can now plot the graph on your own. Let’s now move on to finding solution of linear inequality using graphical representation
Solution of inequality
Understand linear equation line basically divides the line into two parts, one part is the solution plane and the other part is non-solution plane
For example consider the three cases below
CASE A – Linear Expression
(a) 2x +4y = 8
This is a simple linear equation whose all solution points lie on the line
CASE B – Expression with “>” sign
Let us now try to find the solution of inequality with “>” sign
2x + 4y > 8
Step 01
First plot the expression 2x+4y =8 in the graphical figure which will divide the graph into two planes (upper plane and lower plane)
Step 02
One of the entire plane is the solution of inequality equation 2x+4y >8
Let’s check for lower plane where point O (0,0) lies
Putting the value (0,0) in 2x+4y >8
2(0) + 4(0) >8
0 > 8
Condition is not satisfied, hence lower plane is not the solution
Step 03
lets now check if upper plane is the solution
Point (4, 2) lies in upper plane, putting this value in inequality 2x+4y>8, we get
2(4) + 4(2) >8
8 + 8 >8
16 > 8
Condition is satisfied, hence all the upper plane is part of the solution of inequality 2x+ 4y >8
Note: the points on the line 2x + 4y=8 is not the solution of inequality 2x+4y >8 because the solution is the point which lies above the line (due to presence of “>” sign)
Case C – Expression with “>” sign
Let us find solution of inequality with ” < ” sign
2x + 4y < 8
Step 01
Draw the expression 2x+4y = 8 in the graph chart
Step 02
The expression 2x+4y = 8 divides the chart into two parts, upper and lower.
One of the part is the solution of inequality 2x + 4y< 8
Lets check for the lower plane in the graph.
The point (0, 0) lies in the lower plane, check if it satisfies the inequality condition
2(0) +4(0) < 8
0 < 8
The condition is satisfied, hence lower plane is the solution of given inequality. For reference, see the graph below.
Summary
Till now we have studied 3 cases
(A) 2x + 4y = 8
Points on the line are the solution
(B) 2x + 4y > 8
Upper plane of the line is the solution
(C) 2x + 4y < 8
Lower plane of the line is the solution
Let us now solve some questions to further clear our concept of Grade 11 Inequality.
Inequality Math Questions
(01) Solve the following inequality using two dimensional graph
3y – 5x < 30
Step 01
First plot the line of the expression 3y – 5x =30
(a) put y=0, we get x= -6
(b) put x=0, we get y= 10
Plotting (-6,0) and (0,10) point in the graph and joining it with a line
You can see that the line 3y – 5x=30 divides the graph into two plane (plane 1 and plane 2). One of the plane is the solution of the inequality 3y – 5x < 30
Step 02
Take a point in 1st plane and check if it satisfies the condition.
Point (2,2) lie on the first plane, putting the values in inequality equation
3(2) -5(2) <30
6-10 < 30
-4 < 30
The condition is satisfied, hence all the points on plane 1 are the solution of inequality
(02) Solve the inequality using graphs
2x + y ≥ 6
Solution
Step 01: Draw the graphical representation of 2x+y = 6
(a) put x = 0, we get y=6
(b) put y=0, we get x=3
Plot the points (3, 0) and (0, 6) in the graph and join the with straight line
The line divide the graph into two parts ( Part 01 and Part 02). One of the part is our solution.
Step 02
Let’s check the part 01 for the inequality 2x + y ≥ 6
Point (6,0) lies in Part 01 of the given figure, let’s see if it satisfies the inequality condition
2(6) ≥ 6
12 ≥ 6
The condition is satisfied which means that all the points lying in the Part 01 of the figure is the solution,
Note: The points on the line 2x+y = 6 is also part of the solution because the inequality involves (2x + y ≥ 6) equal to sign in the equation
