How to Find the Solution of the Inequality
HOW TO SOLVE SYSTEMS OF LINEAR INEQUALITIES IN ONE VARIABLE
How to solve systems of linear inequalities in one variable :
In this section, we will learn, how to solve the system of linear inequalities in one variable.
Solving System Linear Inequalities in One Variable - Steps
Step 1 :
Solve both the given inequalities and find the solution sets. Also sketch the graphs of the solution sets.
Step 2 :
Combine the graphs of the solution sets of both the inequalities. Identify the common region of the solution sets.
Step 2 :
The values in the common region of the solution sets is the final solution set of the given system of inequalities. Because the values of the variable in the common region will satisfy both the inequalities in the system.
Solving System of Linear Inequalities in One Variable - Examples
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]
After having gone through the stuff given above, we hope that the students would have understood, how to solve systems of linear inequalities in one variable.
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How to Find the Solution of the Inequality
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