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, +∞)

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, +∞)

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