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Inequalities

Inequalities are statements where two quantities are not equal, but a relationship exists between them. It indicates that one quantity is less than (less than or equal to) or more than (more than or equal to) another quantity.

Inequalities may be of one or more than one variable.

 

The simplest linear inequation we can find is of the form Description: 25819.png 

 

The solution space for an inequality is the set of all values that satisfy the inequality.

 

The solution space for the inequality Description: 25825.pngare all points to the left of the y-axis and the solution space for the inequality Description: 25831.png are all points to the right of the y-axis as shown below.

 

Similarly, the solution space for the inequality Description: 25878.pngwill be all points that are below the x-axis and the solution space for the inequality Description: 25884.png will be all points that are above the x-axis.

 

To find the solution space for any inequality, first we consider the inequality as a linear equation and draw a line corresponding to that equation. Then, substitute zero in place of the variables in the inequality. If the inequality holds good, then shade the side of the line having the origin, else shade the other side.

 

Let us learn how to find the solution space for an inequality in one variable.

Graph of an Inequation

The following steps are required for the graph of an inequation Description: 25903.png

 

Step 1: Write down the inequality as an equality

For example: Write ax + by = c for the inequality Description: 25915.png or for Description: 25925.png

 

Step 2: Form a table of values of the equation ax + by = c

 

Step 3: Plot the points of the table obtained in Step 2 in XOY plane and join them. It should be a straight line.

 

Step 4: Determine the region in which the given inequation is satisfied.

 

Step 5: Shade the desired region.
 

Example
Show the solution space for the inequality y > 3.
Solution
First let us consider the given inequality as a linear equation.
 
Description: 35868.png 
 
Hence, we have the equation y = 3.
Now, let us draw a line corresponding to this equation on the co-ordinate plane.
Then, substitute zero in place of the variable in the given inequality.
The inequality given to us is y > 3.
Substituting 0 in place of y, we get 0 > 3.
But this inequality is not true. Hence, we have to shade the region that does not contain the origin.
The shaded region is the solution space.
If the inequality contains 2 variables, then it is known as a linear inequality in 2 variables.
For example, 2x + 3y < 6 is a linear inequality in 2 variables.
Let us learn how to find the solution space for an inequality in 2 variables with an example.

 

Example
Show the solution space for the inequality 2 x + 3y < 6.
Solution
First consider the given inequality as a linear equation by substituting the inequality sign with an equality sign. By changing the inequality sign to an equality sign, we get the equation
2x + 3y = 6.
Now, substitute any two convenient values for x and y so that we get two points. For example, in the linear equation 2x + 3y = 6, when x = 0, y = 2 and when y = 0, x = 3. Hence, we get 2 points
(0, 2) and (3, 0).
Plot these points on the co-ordinate plane and join them to get the line corresponding to the linear equation.
Then, substitute zero in place of the variables in the given inequality. Substituting 0 in place of x and y, we get 0 < 6 which is true. Hence, shade the region on the side that contains the origin.
Description: 35887.png 
If we are asked to find the solution space for more than 1 inequality, then we do the above procedure to each of the given inequalities on the same co-ordinate plane. The region that is common to all the inequalities is the solution space.

 

Example
 Show the solution space for the following linear inequalities:
Description: 26475.png 
Solution
Let us first draw the line x + 4y = 12.
Description: 36231.png 
Here, when x = 0, y = 3 and when y = 0, x = 12.
We get the points (0, 3) and (12, 0).
Consider the line 2x + 5y = 20. Here, when x = 0, y = 4 and when y = 0, x = 10.
We get the points (0, 4) and (10, 0).
y = 0 is the x-axis and henceDescription: 23401.pngindicates all points above the x-axis.
x = 1 is the line parallel to y-axis at x = 1 and x = 8 is the line parallel to y-axis at x = 8.
Description: 23407.pngis the region between these two lines.

Points to Remember

  • Adding or subtracting the same number on both sides of an inequality produces an equivalent inequality.
     
    If a, b, c are three real numbers and if a > b, then
    a + c > b + c and a - c > b - c.
  • Multiplying or dividing the same positive number on both sides of an inequality produces an equivalent inequality.
     
    If a, b, c are three real numbers and if a > b and c > 0, then
    ac > bc and Description: 26521.png
  • Multiplying or dividing the same negative number on both sides of an inequality produces an inequality with the sign of the inequality reversed.
     
    If a, b, c are three real numbers and if a > b and c < 0, then
    ac < bc and Description: 33983.png
  • If a, b, c, d are four real numbers and if
    a > b and c > d then a + c > b + d
    a < b and c < d then a + c < b + d
  • The square of any real number (positive or negative) is always > 0.

Method of Solving Linear Inequations

Consider a linear inequation, say, ax + b > c, where a > 0 and a, b, c are real numbers.

 

Given: ax + b > c

 

Adding (-b) to both sides:

 

ax + b + (-b) > c + (-b)  ax > c - b

 

Dividing both sides by a:

 

Description: 23431.png Description: 23437.png 
 

Example
A company produces two products x and y, each of which requires processing in two machines. The first machine can be used at most for 100 hours; the second machine can be used at most for 50 hours. The product x requires 2 hours on machine one and 1 hour on machine two. The product B requires 1 hour on machine one and 2 hours on machine two. Express the above situation using linear inequalities.
Solution
Let the company produces p numbers of product x and q numbers of product y. As each of the product x requires 2 hours in machine one and 1 hour in machine two, p numbers of product x require 2p hours in machine one and p hours in machine two. Similarly, q numbers of product y require q hours in machine one and 2q hours in machine two. But machine one can be used for 100 hours and machine two for 50 hours. Hence, 2p + q cannot exceed 100 and p + 2q cannot exceed 50.
 
Therefore, 2p + q ≤ 100 and p + 2q ≤ 50 are the required inequalities.

 

Example
Solve 18 < 3x - 2, where x is an integer.
Solution
Given 18 < 3x - 2
 
Adding 2 on both sides, we get
 
18 + 2 < 3x - 2 + 2 ⇒ 20 < 3x
 
Dividing both sides by 3, we get
 
Description: 23461.png Description: 23467.png 
 
⇒ x = 7, 8, 9... Solution set is {7, 8, 9...}.





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