SOLVING ABSOLUTE VALUE INEQUALITIES
About "Solving Absolute Value Inequalities"
Solving Absolute Value Inequalities :
In this section, we are going to learn, how to solve absolute value inequalities.
Solving Absolute Value Inequalities
The general form of an absolute value inequality is
|ax + b| ≤ k
or
|ax + b| ≥ k
Method 1 : (Less Than or Equal to)
Solve the absolute value inequality given below
|x + 2| ≤ 3
Let us graph the solution of the first branch x ≤ 1
Let us graph the solution of the second branch x ≥ -5
If we combine the above two graphs, we will get a graph as given below.
From the above graph, the solution for |x + 2| ≤ 3 is
-5 ≤ x ≤ 1
Method 2 : (Greater Than or Equal to)
Solve the absolute value inequality given below
|x - 3| ≥ 1
Solution :
We can solve the absolute value inequality |x - 3| ≥ 1 as shown below.
Let us graph the solution of the first branch x ≥ 4
Let us graph the solution of the second branch x ≤ 2
If we combine the above two graphs, we will get a graph as given below.
From the above graph, the solution for |x - 3| ≥ 1 is
(-∞, 2] U [3, +∞)
Some More Examples
Example 1 :
Solve the absolute value inequality :
|2x + 1| ≤ 5
Solution :
Solve :
2x + 1 ≤ 5 or 2x + 1 ≥ -5
2x ≤ 4 or 2x ≥ -6
x ≤ 2 or x ≥ -3
Hence, the solution is
-3 ≤ x ≤ 2
Example 2 :
Solve the absolute value inequality :
|3x + 5| ≥ 7
Solution :
Solve :
3x + 5 ≥ 7 or 3x + 5 ≤ -7
3x ≥ 2 or 3x ≤ -12
x ≥ 2/3 or x ≤ -4
Hence the solution is
(-∞, -4] U [2/3, +∞)
Example 3 :
Solve the absolute value inequality :
|x - 1| + 2 ≤ 5
Solution :
Solve :
|x - 1| + 2 ≤ 5
Subtract 2 from each side.
|x - 1| ≤ 3
x - 1 ≤ 3 or x - 1 ≥ -3
x ≤ 4 or x ≥ -2
Hence the solution is
-2 ≤ x ≤ 4
Example 4 :
Solve the absolute value inequality :
|2x - 3| - 5 ≥ 7
Solution :
Solve :
|2x - 3| - 5 ≥ 7
Add 5 to each side.
|2x - 3| ≥ 12
2x - 3 ≥ 12 or 2x - 3 ≤ -12
2x ≥ 15 or 2x ≤ -9
x ≥ 15/2 or x ≤ -9/2
Hence the solution is
(-∞, -9/2] U [15/2, +∞)
Example 5 :
Solve the absolute value inequality :
2|x + 1| ≤ 6
Solution :
Solve :
2|x + 1| ≤ 6
Divide each side by 2.
|x + 1| ≤ 3
x + 1 ≤ 3 or x + 1 ≥ -3
x ≤ 2 or x ≥ -4
Hence the solution is
-4 ≤ x ≤ 2
Example 6 :
Solve the absolute value inequality :
5|x - 3| ≥ 15
Solution :
Solve :
5|x - 3| ≥ 15
Divide each side by 5.
|x - 3| ≥ 3
x - 3 ≥ 3 or x - 3 ≤ -3
x ≥ 6 or x ≤ 0
Hence the solution is
(-∞, 0] U [6, +∞)
Example 7 :
Solve the absolute value inequality :
2|x + 3| + 5 ≤ 13
Solution :
Solve :
2|x + 3| + 5 ≤ 13
Subtract 5 from each side.
2|x + 3| ≤ 8
Divide each side by 2.
|x + 3| ≤ 4
x + 3 ≤ 4 or x + 3 ≥ -4
x ≤ 1 or x ≥ -7
Hence the solution is
-7 ≤ x ≤ 1
Example 8 :
Solve the absolute value inequality :
5|x +7| - 2 ≥ 18
Solution :
Solve :
5|x +7| - 2 ≥ 18
Add 2 to each side.
5|x +7| ≥ 20
Divide each side by 5.
|x +7| ≥ 4
x + 7 ≥ 4 or x + 7 ≤ -4
x ≥ -3 or x ≤ -11
Hence the solution is
(-∞, -11] U [-3, +∞)
Example 9 :
Solve the absolute value inequality :
|x + 3| < 13
Solution :
Solve :
|x + 3| < 13
x + 3 < 13 or x + 3 > -13
x < 10 or x > -16
Hence the solution is
-16 < x < 10
Example 10 :
Solve the absolute value inequality :
|x +7| > 18
Solution :
Solve :
|x +7| > 18
x + 7 > 18 or x + 7 < -18
x > 11 or x < -25
Hence the solution is
(-∞, -25) U (11, +∞)
After having gone through the stuff given above, we hope that the students would have understood, "Solving Absolute Value Inequalities".
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