HOW TO SOLVE SYSTEMS OF LINEAR INEQUALITIES IN ONE VARIABLE

Example 1 :

Solve the following system of linear inequalities.

(5x/4) + (3x/8)  >  39/8

(2x - 1)/12 - (x - 1)/3  <  (3x + 1)/4

Solution :

Solving the first inequality : 

(5x/4) + (3x/8)  <  39/8

10x/8 + 3x/8  <  39/8

(10x + 3x)/8  <  39/8

13x/8  <  39/8

Multiply each side by 8.

13x  <  39

Divide each side by 13. 

x  <  3

Solution set for the first inequality is

(-∞, 3)

Sketch the graph :

Solving the second inequality :

(2x - 1)/12 - (x - 1)/3  <  (3x + 1)/4

(2x - 1)/12 - 4(x - 1)/12  <  (3x + 1)/4

[(2x - 1) - 4(x - 1)]/12  <  (3x + 1)/4

[2x - 1 - 4x + 4]/12  <  (3x + 1)/4

(-2x + 3)/12  <  (3x + 1)/4

Multiply each side by 12. 

(-2x + 3)  <  3(3x + 1)

-2x + 3  <  9x + 3

Add 2x to each side.  

3  <  11x + 3

Subtract 3 from each side.

0  <  11x

Divide each side by 11.

0  <  x

So, the solution set for the second inequality is

(0, ∞)

Sketch the graph :

Combine the graphs of the solution sets of the first and second inequalities. 

In the above graph, the common region found in the solution sets of the first and second inequalities is 

(0, 3)

Therefore, the solution set for the given system of inequalities is

(0, 3)

Example 2 :

Solve the following system of linear inequalities 

x / (2x + 1)  ≥  1 / 4

6x / (4x - 1)  <  1 / 2 

Solution :

Solving the first inequality :

x / (2x + 1)  ≥  1 / 4

Multiply each side by 4(2x + 1).

4x  ≥  2x + 1

Subtract 2x from each side. 

2x  ≥  1

Divide each side by 2.

x  ≥  1 / 2

So, solution set for the first inequality is

[1/2, ∞)

Sketch the graph :

Solving the second inequality :

6x / (4x - 1)  <  1 / 2

Multiply each side by 2(4x - 1).

12x  <  4x - 1

Subtract 4x from each side. 

8x  <  - 1

Divide each side by 8.

x  <  - 1 / 8

So, the solution set for the second inequality is

(- ∞, - 0.125)

Sketch the graph :

Combine the graphs of the solution sets of the first and second inequalities. 

In the above graph, there is common region found in the solution sets of the first and second inequalities. 

Therefore, no solution for the given system of inequalities. 

Example 3 :

Solve the following system of linear inequalities 

3x - 6  ≥  0,  4x - 10  ≤  6

Solution :

Solving the first inequality : 

3x - 6  ≥  0

Add 6 to each side. 

3x  ≥  6

Divide each side by 3. 

x  ≥  2

So, the solution set for the first inequality is

[2, ∞)

Sketch the graph :

Solving the second inequality : 

4x - 10  ≤  6

Add 10 to each side. 

4x  ≤  16

Divide each side by 4. 

x  ≤  4

So, the solution set for the second inequality is

(-∞, 4]

Sketch the graph :

Combine the graphs of the solution sets of the first and second inequalities. 

In the above graph, the common region found in the solution sets of the first and second inequalities is 

[2, 4]

Therefore, the solution set for the given system of inequalities is

[2, 4]

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