SOLVE LINEAR INEQUALITIES

A statement involving variable or variables and the sign of inequality like  <,  >,  ≤  or  ≥  is called an inequality.

Here we are going to see how to solve linear inequalities.

• Solving linear inequalities in one variable
• Solving linear inequalities in two variables

Linear Inequalities in One Variable

Let a be a non zero real numbers and x be a variable.

Then, the inequality of the form

ax + b  <  0

ax + b  ≤  0

ax + b  >  0

ax + b  ≥  0

are known as linear inequalities in one variable.

Example :

Solve the following linear inequality.

2x - 4  ≤  0

Solution :

2x - 4  ≤  0

Add 4 to each side.

2x - 4 + 4  ≤   0 + 4

2x  ≤  4

Divide each side by 2. 

x  ≤  2

x  ≤  2

By graphing this in the number line, we can split this as two intervals. (-∞, 2] and [2, ∞).

Now we need to select one of the values from the above intervals and apply those values instead of x in the given question 2x - 4  ≤  0.

The values from which interval makes the given inequality true is the solution set.

Therefore, (-∞, 2] is the solution set.

Linear Inequalities in Two Variables

Let a, b, c be real numbers. 

Then, the equation ax + by + x = 0 is called a linear equation in two variables.

Whereas the inequalities 

ax + by  <  c

ax + by  ≤  c

ax + by  >  c

ax + by  ≥  c

are known as linear inequalities in two variables.

Example :

Solve the following inequality graphically

2x - y  ≥  1

Solution :

Consider the linear inequality as if it were a linear equation

2x - y  =  1 -----(1)

Find x-intercept : 

Substitute 0 for y in (1)

(1)-----> 2x  - 0  =  1

2x  =  1

Divide each side by 2.

x  =  0.5

So, x-intercept is (0.5, 0)

Find y-intercept : 

Substitute 0 for x in (1)

(1)-----> 2(0)  - y  =  1

-  y  =  1

Multiply each side by (-1).

y  =  - 1

So, y-intercept is (0, -1).

Using the x-intercept (0.5, 0) and y-intercept (0, -1), we can draw a straight  as shown below. 

Now we need to take a point on either sides of the line shown above.

Take the two points (-2, 3) and and (2, -2). 

Case 1 :

Substitute the point (-2, 3) into the inequality. 

2(-2) - 3  ≥  1  ?

- 4 - 3  ≥  1  ?

- 7  ≥  1  ?

- 7 can never be greater than 1

So, the above statement is false.

Hence, the point (-2, 3) does not satisfy the inequality. 

Case 2 :

Substitute the point (2, -2) into the inequality. 

2(2) - (-2)  ≥  1  ?

4 + 2  ≥  1  ?

6  ≥  1  ?

6 is greater than 1

So, the above statement is true.

Hence, the point (-2, 2) satisfies the inequality. 

Shade the Solution Region : 

Now, we have to shade the region from where the point (2, -2) has been taken.

That is the solution region. 

The solution region is shown in the graph below.

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