SOLVING ABSOLUTE VALUE INEQUALITIES WORKSHEET

About "Solving Absolute Value Inequalities Worksheet"

Solving Absolute Value Inequalities Worksheet :

Worksheet given in this section is much useful to the students who would like to practice problems on solving absolute value inequalities worksheet. 

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Solving Absolute Value Inequalities Worksheet - Problems

Problem 1 :

Solve the absolute value inequality : 

|2x + 1|  ≤  5

Problem 2 :

Solve the absolute value inequality :

|3x + 5|  ≥  7

Problem 3 :

Solve the absolute value inequality :

|x - 1| + 2  ≤  5

Problem 4 :

Solve the absolute value inequality :

|2x - 3| - 5    7

Problem 5 :

Solve the absolute value inequality :

2|x + 1|  ≤  6

Problem 6 :

Solve the absolute value inequality :

5|x - 3|  ≥  15

Problem 7 :

Solve the absolute value inequality :

2|x + 3| + 5    13

Problem 8 :

Solve the absolute value inequality :

5|x +7| - 2  ≥  18

Problem 9 :

Solve the absolute value inequality :

|x + 3|  <  13

Problem 10 :

Solve the absolute value inequality :

|x +7|  >  18

Solving Absolute Value Inequalities Worksheet - Solutions

Problem 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 

Problem 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, +∞)

Problem 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 

Problem 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, +∞)

Problem 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

Problem 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, +∞)

Problem 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

Problem 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, +∞)

Problem 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

Problem 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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