Linear Inequalities Class 11

What Are Linear Inequalities?

Linear Inequalities can be explained as an inequality (represented by the symbols of inequality) that holds a linear function. A linear function can be described as any function whose graph is a straight line. Now, if you are wondering what inequality means in the field of Mathematics. Here is your answer. When two real numbers or two algebraic expressions are represented with symbols like <, > or ≤, ≥  they can be called an inequality. For example, 3x<20, or 4x+y>12

Meaning Of The Symbols Used In Inequalities:

• The symbol < means ‘less than’ and ≤ means less than or equal to.

• The symbol > means ‘greater than’ and ≥ means greater than or equal to. 

• The symbol ≠ means the qualities on either side of it are not equal.

Properties Of Inequalities

• So far whatever properties that you have learned to solve linear equations will be applied in solving inequalities too. The only difference will be when you perform multiplication or division by a negative, you have to reverse the inequality sign as well. 

Types Of Inequalities

There are Four Types of Inequalities, They are:

• Strict: The inequalities that have < or > symbol between the L.H.S and R.H.S.

• Slack: The inequalities that have ≤ or ≥  symbol between the L.H.S and R.H.S.

• Linear: The inequalities that have a degree 1. Example, 5x + 2y>10

• Quadratic: The inequalities that have a degree 2. Example, 5x2 + 2y>10

Linear Inequalities In One Variable

A linear equation in one variable holds only one variable and whose highest index of power is 1. Here are a few examples of linear inequation in one variable:

9x - 2 <0

5x + 27>0

20x - 7 ≥ 0

Linear Inequalities In Two Variable

A linear equation in one variable holds only two variables.

For example, 20x - 7y ≥ 0

Steps To Solve Linear Inequalities In One Variable

In order to solve linear inequalities in one variable, you must follow a few steps. 

Step 1) first, obtain the linear inequation.

Step 2) In this step, drag all the terms containing variables to one side and those with constant to the other side. 

Step 3) Now, simplify the final equation.

Step 4) In this step you have to divide the coefficient of the variable on both sides but you need to remember that if the coefficient is positive then the direction of the inequality will not change. If the coefficient is negative then the direction of the inequality will change.

Step 5) In this final step you have to put the result on a number line and thus get the solution set in interval form. 

Here is an Example For You.

Solve: x-2>2x+5

Solution: we start by subtracting x from both the side

  x-2>2x+5   x-2-x>2x+5-x = -2>x+5

Now by subtracting 5 from both the sides, we get:

-2>x+5 - 7>x

Graphing Linear Inequalities 

While graphing there are a few points that we must remember, they are:

If the inequality involves either < or > then the lines on the graph will be dotted to indicate that they don’t belong from the solution set. If they include \[\leq\] or \[\geq\] then the lines will be dark indicating that they belong to the solution set. 

How To Find Solution Graphically For Linear Inequalities In Two Variables

To represent linear inequalities on a graph, there are few steps that are to be followed in order to avoid any mistake. 

Step 1) First of all, draw a graph of the equation but remember to replace the inequality sign with an equal sign.

Step 2) Use a dashed line if the inequality involves either < or >. Use a dotted line if they include ≤ or ≥ .

Step 3) If the line itself constitutes a part of the solution, use a solid line.

Step 4) Pick a point lying in one of the half-planes then substitute the values of x and y into the provided inequality.

Step 5) The graph of the inequality will include the half-plane containing the test points if the equality is satisfied, otherwise the half-plane won’t be containing the test points. 

Solved Examples

Example 1) Solve 30 x < 200 if (i) x is a natural number, (ii)  x is an integer. 

Solution 1) We are given 30 x < 200

 or \[\frac{30x}{30} < \frac{200}{30}\] (Rule 2), i.e., x < 20 / 3.

(i) When x is a natural number, in this case, the following values of  x make the statement true. 

                                                   1, 2, 3, 4, 5, 6. 

The solution set of the inequality is {1,2,3,4,5,6}. 

(ii) When x is an integer, the solutions of the given inequality are .., – 3, –2, –1, 0, 1, 2, 3, 4, 5, 6 The solution set of the inequality is {...,–3, –2,–1, 0, 1, 2, 3, 4, 5, 6}

Example 2)  Solve 4x + 3 < 6x +7.

Solution 2) We have, 4x + 3 < 6x + 7

or 4x – 6x < 6x + 4 – 6x

or – 2x < 4 orx > – 2

i.e., all the real numbers that are greater than –2, are the solutions of the given inequality. Hence, the solution set is (–2, ∞).

Example 3)  \[\frac{5-2x}{3}\] ≤  \[\frac{x}{6} -5\]

Solution 3) we have   \[\frac{5-2x}{3}\] ≤  \[\frac{x}{6} -5\]

Or  2 ( 5 - 2x) ≤ x - 30

Or 10 - 4x ≤ x - 30

Or – 5x ≤ – 40, i.e., x ≥ 8

Therefore, all real numbers x that is greater than or equal to 8 are the solutions of the given inequality, i.e.,x ∈ [8, ∞).

Example 4)  Solve 7x + 3 < 5x + 9 then show the graph of the solutions on the number line.

Solution 4) We have 7x + 3 < 5x + 9 

Or 2x < 6 or x < 3 

The graphical representation of the solutions are given below

                         (Image to be added soon)