# SOLVING ABSOLUTE VALUE INEQUALITIES WORKSHEET

Solving Absolute Value Inequalities Worksheet :

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

Before look at the worksheet, if you would like to know the basic stuff about solving absolute value inequalities,

## 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

|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

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, +∞) After having gone through the stuff given above, we hope that the students would have understood, how to solve absolute value inequalities.

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