# Solving Absolute Value Inequalities Worksheet, Definition, and Examples

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### Key Points about Solving Absolute Value Inequalities

- Absolute value inequalities involve the absolute value of a variable and can be used in a variety of contexts.
- To solve absolute value inequalities, it is important to understand the definition of absolute value and consider both the positive and negative solutions.
- Methods for solving absolute value inequalities include using a number line and algebraic methods, such as simplifying the inequality using properties of absolute value.

## What are Absolute Value Inequalities?

Absolute value inequalities are a type of inequality that involves the absolute value of a variable. These inequalities can be tricky to solve, but with the right approach, they can be simplified and solved. Absolute value inequalities can be used in a variety of contexts, from solving mathematical problems to real-world applications.

To solve absolute value inequalities, it is important to understand the definition of absolute value. The absolute value of a number is its distance from zero on the number line. This means that the absolute value of a negative number is the same as the absolute value of its positive counterpart. When solving absolute value inequalities, it is important to consider both the positive and negative solutions.

One common method for solving absolute value inequalities is to use a number line. By plotting the solutions on a number line, it is easier to see which values satisfy the inequality. In addition to using a number line, algebraic methods can also be used to solve absolute value inequalities. By using the properties of absolute value, such as the fact that |a| = |-a|, it is possible to simplify the inequality and solve for the variable.

**Common Core Standard:**

**How to Solve Absolute Value Inequalities**

Absolute value inequalities are used to express the range of values that satisfy a given condition. To solve absolute value inequalities, one needs to isolate the absolute value, split the inequality, and solve the resulting equations.

**Isolate the Absolute Value**

The first step in solving absolute value inequalities is to isolate the absolute value. This can be done by adding or subtracting the same value from both sides of the inequality. For example, consider the following inequality:

|2x – 3| > 5

To isolate the absolute value, one needs to add 3 to both sides of the inequality. This gives:

|2x – 3| + 3 > 5 + 3

Simplifying the left-hand side gives:

|2x – 3 + 3| > 8

Which simplifies further to:

|2x| > 8

**Split the Inequality**

The second step is to split the inequality into two cases, one for when the expression inside the absolute value is positive, and one for when it is negative. For example, using the same inequality as above:

|2x| > 8

One needs to consider two cases:

- 2x > 8
- 2x < -8

**Solving the Resulting Equations**

The final step is to solve the resulting equations for each case. For the first case, 2x > 8, one needs to divide both sides by 2, which gives x > 4. For the second case, 2x < -8, one needs to divide both sides by 2 and flip the inequality, which gives x < -4.

Therefore, the solution to the absolute value inequality |2x – 3| > 5 is x < -4 or x > 4.

In summary, solving absolute value inequalities involves isolating the absolute value, splitting the inequality into two cases, and solving the resulting equations. By following these steps, one can determine the range of values that satisfy the given condition.

**What is the Definition of Absolute Value Inequalities?**

Absolute value inequalities are inequalities that contain an absolute value expression. Absolute value is the distance of a number from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

To solve absolute value inequalities, the first step is to isolate the absolute value expression. If the inequality is of the form |x| < a, where a is a positive number, then the solution set is all the values of x that are within a distance of a units from zero. This solution set can be expressed in interval notation as (-a, a).

If the inequality is of the form |x| > a, where a is a positive number, then the solution set is all the values of x that are outside a distance of a units from zero. This solution set can be expressed in interval notation as (-∞, -a) ∪ (a, ∞).

If the inequality is of the form |x| ≤ a, where a is a positive number, then the solution set is all the values of x that are within a distance of a units from zero, including zero. This solution set can be expressed in interval notation as [-a, a].

If the inequality is of the form |x| ≥ a, where a is a positive number, then the solution set is all the values of x that are outside a distance of a units from zero, including zero. This solution set can be expressed in interval notation as (-∞, -a] ∪ [a, ∞).

It is important to remember that when solving absolute value inequalities, the solution set may contain more than one interval. Therefore, it is necessary to check each interval to ensure that it satisfies the original inequality.

**Absolute Value Inequalities Examples**

Absolute value inequalities are inequalities in algebra that involve algebraic expressions with absolute value symbols and inequality symbols. They are solved in a similar way to absolute value equations. The absolute value inequalities can be solved by isolating the absolute value expression and then breaking it into two separate inequalities.

Here are some examples of absolute value inequalities:

**Example 1**

Solve the inequality: |2x – 5| ≤ 3

**Solution:**

To solve this inequality, we need to break it into two separate inequalities:

2x – 5 ≤ 3 and -(2x – 5) ≤ 3

Solving the first inequality, we get:

2x – 5 ≤ 3

2x ≤ 8

x ≤ 4

Solving the second inequality, we get:

-(2x – 5) ≤ 3

-2x + 5 ≤ 3

-2x ≤ -2

x ≥ 1

Therefore, the solution to the inequality is 1 ≤ x ≤ 4.

**Example 2**

Solve the inequality: |3x + 1| > 7

**Solution:**

To solve this inequality, we need to break it into two separate inequalities:

3x + 1 > 7 and -(3x + 1) > 7

Solving the first inequality, we get:

3x + 1 > 7

3x > 6

x > 2

Solving the second inequality, we get:

-(3x + 1) > 7

-3x – 1 > 7

-3x > 8

x < -8/3

Therefore, the solution to the inequality is x < -8/3 or x > 2.

**Example 3**

Solve the inequality: |4x – 1| ≥ 5

**Solution:**

To solve this inequality, we need to break it into two separate inequalities:

4x – 1 ≥ 5 and -(4x – 1) ≥ 5

Solving the first inequality, we get:

4x – 1 ≥ 5

4x ≥ 6

x ≥ 3/2

Solving the second inequality, we get:

-(4x – 1) ≥ 5

-4x + 1 ≥ 5

-4x ≥ 4

x ≤ -1

Therefore, the solution to the inequality is x ≤ -1 or x ≥ 3/2.

**Graphing Absolute Value Inequalities**

Graphing absolute value inequalities involves plotting the solutions to the inequality on a number line. The first step is to isolate the absolute value expression on one side of the inequality. Then, split the inequality into two parts: one for when the expression inside the absolute value is positive and one for when it is negative.

For example, consider the inequality |x – 2| < 5. To solve this inequality, first isolate the absolute value expression by subtracting 2 from both sides to obtain |x – 2| – 2 < 5 – 2. This simplifies to |x – 2| < 3.

Next, split the inequality into two parts: x – 2 < 3 and x – 2 > -3. Solving for x in both parts gives x < 5 and x > -1. The solution set is the intersection of these two parts, which is -1 < x < 5.

To graph the solution set on a number line, draw a line with an open circle at -1 and an open circle at 5. Shade the region between these two points to represent the solution set.

It is important to note that when the inequality involves a less than or equal to or greater than or equal to symbol, the open circles on the number line should be replaced with closed circles.

In summary, to graph absolute value inequalities, isolate the absolute value expression, split the inequality into two parts, solve for x in each part, and graph the solution set on a number line.

**Properties of Absolute Value Inequalities**

Absolute value inequalities are inequalities that contain absolute values. These types of inequalities are used to solve problems in various fields such as engineering, physics, and economics. The properties of absolute value inequalities are important to understand in order to solve them effectively.

**Property 1: Definition of Absolute Value**

The absolute value of a number is always positive or zero. For example, the absolute value of 3 is 3, and the absolute value of -3 is also 3. This property is important to remember when solving absolute value inequalities because it helps to determine the possible solutions.

**Property 2: Solving Absolute Value Inequalities**

To solve absolute value inequalities, it is important to first isolate the absolute value expression. Once the absolute value expression is isolated, the inequality can be split into two cases: one where the expression is positive, and one where it is negative. This is because the absolute value of a number can be positive or negative.

**Property 3: Graphical Representation**

Absolute value inequalities can also be represented graphically on a number line. This helps to visualize the possible solutions and determine the intervals where the inequality is true. For example, if the absolute value inequality is |x-3| > 4, the solutions lie outside the interval (-1, 7) on the number line.

**Property 4: Multiplication and Division**

When multiplying or dividing both sides of an absolute value inequality by a negative number, the inequality sign must be reversed. This is because multiplying or dividing by a negative number changes the direction of the inequality.

**Property 5: Absolute Value Inequalities with Multiple Absolute Values**

Absolute value inequalities can also contain multiple absolute values. In this case, the inequality can be solved by isolating each absolute value expression and splitting the inequality into multiple cases.

Overall, understanding the properties of absolute value inequalities is crucial for solving them effectively. By applying these properties, one can determine the possible solutions and graphically represent them on a number line.

**What makes an Absolute Value Inequalities No Solution?**

Absolute value inequalities can have a solution, one solution, or no solution. An absolute value inequality has no solution when the absolute value expression is always greater than or equal to zero. This means that the inequality cannot be satisfied by any value of the variable.

For example, consider the inequality |x + 2| < -3. Since the absolute value of any number is always greater than or equal to zero, the left-hand side of this inequality can never be negative. Therefore, there is no value of x that can satisfy the inequality, and it has no solution.

Another way to determine if an absolute value inequality has no solution is to graph the inequality on a number line. If the graph does not intersect with any part of the number line, then there is no solution to the inequality.

It is important to note that an absolute value inequality may have no solution even if the corresponding absolute value equation has a solution. This is because an absolute value inequality involves a range of values, while an absolute value equation involves a specific value.

In summary, an absolute value inequality has no solution when the absolute value expression is always greater than or equal to zero, or when the graph of the inequality does not intersect with any part of the number line.

**Absolute Value Inequalities FAQ**

**How to solve absolute value inequalities with fractions?**

To solve absolute value inequalities with fractions, one needs to isolate the absolute value expression first. After isolating the absolute value expression, one can split the inequality into two cases: one with the positive argument and the other with the negative argument. Then, solve each case separately and combine the solutions.

**What are the steps to solve inequalities with absolute values on both sides?**

The steps to solve inequalities with absolute values on both sides are to isolate the absolute value expression on one side of the inequality, and then split the inequality into two cases. After solving each case, combine the solutions.

**Solving absolute value inequalities with variables on both sides: What is the process?**

To solve absolute value inequalities with variables on both sides, one needs to isolate the absolute value expression on one side of the inequality. Then, split the inequality into two cases: one with the positive argument and the other with the negative argument. After solving each case, combine the solutions.

**How to practice solving absolute value inequalities?**

One can practice solving absolute value inequalities by doing practice problems and exercises. Additionally, textbooks and workbooks on algebra and precalculus often have sections on solving absolute value inequalities.

**What is the significance of no solution in solving absolute value inequalities?**

The significance of no solution in solving absolute value inequalities is that it means there is no value of the variable that satisfies the inequality. This can happen when the inequality is inconsistent, or when the absolute value expression is always positive or always negative.

**What are the key concepts of solving absolute value inequalities according to Khan Academy?**

The key concepts of solving absolute value inequalities according to Khan Academy are to treat the absolute value expression as a single quantity, isolate the absolute value expression, and split the inequality into two cases. After solving each case, combine the solutions.

**What is the concept of absolute value inequalities?**

The concept of absolute value inequalities is that they are inequalities that involve the absolute value of a variable. Absolute value inequalities can be solved by isolating the absolute value expression, splitting the inequality into two cases, and solving each case separately.

**When absolute value inequalities are greater than negative?**

Absolute value inequalities are greater than negative when the absolute value expression is positive. For example, |x| > -3 is true for all values of x.

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