Solving absolute value inequalities worksheet is much useful to the students who would like to practice problems on absolute value functions.

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Now let us look at absolute value inequalities worksheet.

Solve the following absolute value inequalities.

1) |x + 2| ≤ 3

2) |x - 3| ≥ 1

3) |2x + 1| ≤ 5

4) |3x + 5| ≥ 7

5) |x - 1| +2 ≤ 5

6) |2x - 3| - 5 ≥ 7

7) 2|x + 1| ≤ 6

8) 5|x - 3| ≥ 15

9) 2|x + 3| + 5 ≤ 13

10) 5|x +7| - 2 ≥ 18

11) |x + 3| < 13

12) |x +7| > 18

**Problem 1 :**

Solve the absolute value inequality given below

**|x + 2| ****≤ 3**

**Solution :**

Let us see how the above mentioned absolute value inequality can be solved.

The picture given below clearly explains you "How the above inequality can be solved"

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**

**Problem 2 :**

Solve the absolute value inequality given below

**|x - 3| ****≥**** 1**

**Solution :**

Let us see how the above mentioned absolute value inequality can be solved.

The picture given below clearly explains you "How the above inequality can be solved"

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, +****∞)**

**Problem 3 :**

Solve the absolute value inequality given below

**|2x + 1| ****≤**** 5**

**Solution :**

In this problem, we have "less than or equal to " symbol.

So, we have to apply "Method 1"

The two branches are

2x + 1 ≤ 5 (or) 2x + 1 ≥ -5

2x ≤ 4 (or) 2x ≥ -6

x ≤ 2 (or) x ≥ -3

**Hence the solution is**

**-3 ≤ x ****≤ 2**

**Let us look at the next problem on "Absolute value inequalities worksheet"**

**Problem 4 :**

Solve the absolute value inequality given below

**|3x + 5| ****≥**** 7**

**Solution :**

In this problem, we have "greater than or equal to " symbol.

So, we have to apply "Method 2"

The two branches are

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, +****∞)**

**Let us look at the next problem on "Solving absolute value inequalities worksheet"**

**Problem 5 :**

Solve the absolute value inequality given below

**|x - 1| +2 ****≤**** 5**

**Solution :**

First Let us write the given absolute value inequality in standard form.

**|x - 1| + 2 ****≤**** 5 -----------------> ****|x - 1| ****≤**** 3**

In this problem, we have "less than or equal to" symbol.

So, we have to apply "Method 1"

The two branches are

x - 1 ≤ 3 (or) x - 1 ≥ -3

x ≤ 4 (or) x ≥ -2

**Hence the solution is**

**-2 ≤ x ****≤ 4**

**Let us look at the next problem on "Solving absolute value inequalities worksheet"**

**Problem 6 :**

Solve the absolute value inequality given below

**|2x - 3| - 5 ****≥**** 7**

**Solution :**

First Let us write the given absolute value inequality in standard form.

**|2x - 3| - 5 ****≥**** 7 ****-----------------> ****|2x - 3| ****≥**** 12**

In this problem, we have "greater than or equal to " symbol.

So, we have to apply "Method 2"

The two branches are

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, +****∞)**

**Let us look at the next problem on "Solving absolute value inequalities worksheet"**

**Problem 7 :**

Solve the absolute value inequality given below

**2|x + 1| ****≤**** 6**

**Solution :**

First Let us write the given absolute value inequality in standard form.

**2|x + 1| ****≤ 6 **-----------------> **|x + 1| ****≤**** 3**

In this problem, we have "less than or equal to" symbol.

So, we have to apply "Method 1"

The two branches are

x + 1 ≤ 3 (or) x + 1 ≥ -3

x ≤ 2 (or) x ≥ -4

**Hence the solution is**

**-4 ≤ x ****≤ 2**

**Let us look at the next problem on "Solving absolute value inequalities worksheet"**

**Problem 8 :**

Solve the absolute value inequality given below

**5|x - 3| ****≥**** 15**

**Solution :**

First Let us write the given absolute value inequality in standard form.

**5|x - 3| ****≥**** 15 ****-----------------> ****|x - 3| ****≥**** 5**

In this problem, we have "greater than or equal to " symbol.

So, we have to apply "Method 2"

The two branches are

x - 3 ≥ 5 (or) x - 3 ≤ -5

x ≥ 8 (or) x ≤ -2

**Hence the solution is **

**(-****∞, -2] U [8, +****∞)**

**Let us look at the next problem on "Solving absolute value inequalities worksheet"**

**Problem 9 :**

Solve the absolute value inequality given below

**2|x + 3| + 5 ****≤**** 13**

**Solution :**

First Let us write the given absolute value inequality in standard form.

**2|x + 3|+5 ****≤ 13** --------->**2|x + 3| ****≤**** 8 ------->|x+3|**** ≤ 4**

In this problem, we have "less than or equal to" symbol.

So, we have to apply "Method 1"

The two branches are

x + 3 ≤ 4 (or) x + 3 ≥ -4

x ≤ 1 (or) x ≥ -7

**Hence the solution is**

**-7 ≤ x ****≤ 1**

**Let us look at the next problem on "Solving absolute value inequalities worksheet"**

**Problem 10 :**

Solve the absolute value inequality given below

**5|x +7| - 2 ****≥**** 18**

**Solution :**

First Let us write the given absolute value inequality in standard form.

**5|x+7|-2 ****≥**** 18 ****---------> ****5|x+7| ****≥**** 20 --------->|x+7|****≥4**

In this problem, we have "greater than or equal to " symbol.

So, we have to apply "Method 2"

The two branches are

x + 7 ≥ 4 (or) x + 7 ≤ -4

x ≥ -3 (or) x ≤ -11

**Hence the solution is **

**(-****∞, -11] U [-3, +****∞)**

**Let us look at the next problem on "Solving absolute value inequalities worksheet"**

**Problem 11 :**

Solve the absolute value inequality given below

**|x + 3| <**** 13**

**Solution :**

In this problem, we have "less than or equal to" symbol.

So, we have to apply "Method 1"

The two branches are

x + 3 < 13 (or) x + 3 > -13

x < 10 (or) x > -16

**Hence the solution is**

**-16 < x <**** 10**

**Let us look at the next problem on "Solving absolute value inequalities worksheet"**

**Problem 12 :**

Solve the absolute value inequality given below

**|x +7| >**** 18**

**Solution :**

In this problem, we have "greater than or equal to " symbol.

So, we have to apply "Method 2"

The two branches are

x + 7 > 18 (or) x + 7 < -18

x > 11 (or) x < -25

**Hence the solution is **

**(-****∞, -25) U (11, +****∞)**

Apart from the stuff and problems on "Solving absolute value inequalities worksheet, You can also visit the following pages.

**Solving absolute value Equations**

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**Solving absolute value equations worksheet**

**Solving equations with absolute values on both the sides of the equal sign**

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