# Solving absolute value inequalities

##### Intros
###### Lessons
1. $\bullet$ definition of absolute value: $| \heartsuit | =$ distance of $\heartsuit$ from zero
$\bullet$ absolute value inequalities: $| \heartsuit |$ < $a$,
solution:$-a$ < $\heartsuit$ < $a$
$\bullet$ absolute value inequalities: $| \heartsuit |$ > $a$,
solution:$\heartsuit$ < $-a$ $\;or\;$ $\heartsuit$ > $a$
##### Examples
###### Lessons
1. Solving Basic Absolute Value Inequalities
Solve:
1. $|x|$ < $4$
$|x| \leq$ $4$
2. $|x|$ > $4$
$|x| \geq$ $4$
2. Solving Absolute Value Inequalities Involving "less than"
Solve: $|2x-1|$ < $3$
1. Solving Absolute Value Inequalities Involving "greater than"
Solve:
1. $|4x-5|$ > $7$
2. $|x|-5 \geq -1$
2. Multiplying/Dividing an Inequality by a Negative Number
Solve:
1. $|3-2x| \leq 11$
2. $|-\frac{x}{6}+\frac{5}{3}|$ > $2$
3. Given a Pair of Inequalities, Determine the Corresponding Absolute Value Inequality
Determine the absolute value inequality statement that corresponds to each inequality:
1. $-1$ < $x$ < $5$
2. $x \leq-10$ $\;or\;$ $x \geq 2$
4. Recognizing Absolute Value Inequalities with "No Solution" or "All Real Numbers"
Solve:
1. $|x+3|$ < $-5$
2. $|x-4|$ > $-1$

## Introduction to Absolute Value Inequalities

Welcome to our lesson on absolute value inequalities, a fundamental concept in algebra that combines the principles of absolute values and inequalities. To kick off our exploration, we've prepared an introduction video that serves as a crucial starting point for understanding this topic. This video will provide you with a visual and auditory foundation, making the subsequent lessons more accessible. Before we delve into the complexities of inequalities, we'll begin with a comprehensive review of absolute values. This refresher will ensure that everyone has a solid grasp of the basics, which is essential for tackling more advanced concepts. By building on this foundation, we'll gradually introduce the concept of inequalities and how they interact with absolute values. This step-by-step approach will help you develop a deep understanding of absolute value inequalities, enabling you to solve complex problems with confidence.

## Understanding Absolute Values

Absolute value is a fundamental concept in mathematics that represents the distance between a number and zero on a number line, regardless of whether the number is positive or negative. This concept is crucial for understanding various mathematical operations and real-world applications.

To illustrate the concept of absolute value, let's imagine a number line. The absolute value of a number is its distance from zero on this line, always measured in a positive direction. For example, both 4 and -4 are four units away from zero on the number line. Therefore, the absolute value of both 4 and -4 is 4, written as |4| = 4 and |-4| = 4.

This leads us to an important property of absolute values: they are always positive or zero. No matter what number you start with, its absolute value will never be negative. This is because absolute value measures distance, and distance is always a non-negative quantity.

Let's explore more examples to solidify our understanding. The absolute value of 7 is 7 (|7| = 7) because 7 is seven units to the right of zero on the number line. Similarly, the absolute value of -7 is also 7 (|-7| = 7) because -7 is seven units to the left of zero. Notice how the direction doesn't matter; it's only the distance that counts.

To further illustrate why absolute values are always positive, consider this real-life scenario: Imagine you're at home (which we'll consider the zero point on our "real-life number line"). If you walk 5 blocks to school or 5 blocks to the shopping mall, the distance you've traveled is 5 blocks in both cases. It doesn't matter if school is to the east (positive direction) or the mall is to the west (negative direction); the absolute distance remains 5 blocks.

This real-world analogy helps us understand why |-5| and |5| both equal 5. Whether you're moving 5 units in a positive or negative direction on the number line, you're still covering a distance of 5 units from zero.

Absolute value has numerous practical applications. In physics, it's used to calculate the magnitude of vectors. In statistics, it helps measure the spread of data points. In computer programming, it's often used in algorithms for sorting and searching.

Understanding absolute value is also crucial for solving certain types of equations and inequalities. For instance, the equation |x| = 3 has two solutions: x = 3 and x = -3, because both 3 and -3 are three units away from zero on the number line.

In conclusion, absolute value is a powerful mathematical concept that simplifies our understanding of distance on a number line. By always resulting in a non-negative number, it provides a consistent way to measure the magnitude of a quantity, regardless of its direction. Whether you're dealing with positive numbers, negative numbers, or zero, the absolute value gives you a clear picture of a number's distance from the origin, making it an essential tool in various mathematical and real-world applications.

## Properties of Absolute Value

Absolute value in mathematics is a fundamental concept in mathematics with several key properties that are essential for solving various mathematical problems, particularly inequalities. Understanding these properties is crucial for students and professionals alike who work with mathematical concepts. The two most important properties of absolute value are its non-negative nature and the equality of opposites.

The first key property of absolute value is that it is always non-negative. This means that regardless of whether a number is positive, negative, or zero, its absolute value will always be greater than or equal to zero. For example, the absolute value of 5 is 5, the absolute value of -5 is also 5, and the absolute value of 0 is 0. This property is denoted mathematically as |x| 0 for all real numbers x. This non-negative property is particularly useful in situations where we need to measure the magnitude of a quantity without regard to its direction.

The second crucial property is the equality of opposites. This means that the absolute value of a number and its negative are the same. Mathematically, this is expressed as |x| = |-x| for all real numbers x. For instance, |-3| = |3| = 3, and |4| = |-4| = 4. This property highlights the symmetry of the absolute value function and is particularly useful when solving equations involving absolute value in equations.

These properties play a vital role in understanding and solving absolute value in equations. When dealing with inequalities such as |x| < a or |x| > a, where a is a positive real number, the non-negative property ensures that we only consider positive values for a. The equality of opposites property allows us to rewrite these inequalities as compound inequalities. For example, |x| < 3 can be rewritten as -3 < x < 3, encompassing both the positive and negative possibilities that satisfy the original inequality.

Understanding these properties is essential for correctly interpreting and solving problems involving absolute value in mathematics. They form the foundation for more complex mathematical concepts and are widely applied in various fields, including physics, engineering, and computer science. By mastering these properties, students can develop a deeper understanding of absolute value and its applications in real-world scenarios.

## Introduction to Absolute Value Inequalities

Absolute value inequalities are a fundamental concept in algebra that extends our understanding of regular inequalities. While regular inequalities deal with comparing values directly, absolute value inequalities introduce a new layer of complexity by incorporating the absolute value function. This unique mathematical concept plays a crucial role in various fields, from physics to economics, and provides a powerful tool for solving real-world problems.

To grasp the concept of absolute value inequalities, it's essential to first understand what absolute value means. The absolute value of a number is its distance from zero on a number line, regardless of whether it's positive or negative. For example, both 5 and -5 have an absolute value of 5, as they are both 5 units away from zero.

The key difference between regular inequalities and absolute value inequalities lies in how they behave. Regular inequalities follow a straightforward comparison between two values or expressions. For instance, x > 3 simply means that x is greater than 3. However, absolute value inequalities involve the absolute value symbol |x|, which adds a layer of complexity to the interpretation.

Let's consider some simple examples of absolute value inequalities and their graphical representations on a number line:

1. |x| < 3: This inequality represents all numbers whose distance from zero is less than 3. On a number line, this would be represented as all points between -3 and 3, not including -3 and 3 themselves.

2. |x| 4: This inequality includes all numbers whose distance from zero is 4 or greater. On a number line, this would be represented as all points to the left of -4 (including -4) and all points to the right of 4 (including 4).

3. |x - 2| 5: This inequality represents all numbers whose distance from 2 is 5 or less. On a number line, this would be shown as all points between -3 and 7, including -3 and 7.

The graphical representation of absolute value inequalities on a number line is particularly useful for visualizing solutions on number line. For inequalities like |x| < a or |x| > a, the solution is often represented as two separate intervals on the number line. This visual approach helps in understanding the dual nature of absolute value inequalities, where solutions can exist on both sides of zero.

It's important to note that solving absolute value inequalities often requires considering two separate cases due to the nature of absolute value. For example, when solving |x| < 3, we need to consider both x < 3 and x > -3, as the absolute value could represent either a positive or negative number within the given range.

Understanding absolute value inequalities is crucial for advancing in algebra and higher mathematics. These concepts form the foundation for more complex mathematical ideas and have practical applications in various fields. From describing margins of error in scientific measurements to modeling financial scenarios with bounded variables, absolute value inequalities provide a versatile tool for mathematical analysis and problem-solving.

As students progress in their mathematical journey, mastering absolute value inequalities will open doors to more advanced topics such as piecewise functions, optimization problems, and even aspects of calculus. The ability to interpret and solve these inequalities, both algebraically and graphically, is a valuable skill that enhances overall mathematical reasoning and problem-solving capabilities.

## Solving Absolute Value Inequalities

Absolute value inequalities can be challenging for many students, but with a step-by-step approach, they become much more manageable. This guide will walk you through the process of solving both 'less than' and 'greater than' absolute value inequalities, providing algebraic solutions and graphical interpretations.

### Step-by-Step Process for Solving Absolute Value Inequalities

#### Case 1: Less Than (<) Inequalities

1. Isolate the absolute value expression on one side of the inequality.
2. If the inequality is strictly less than (<), keep the inequality sign. If it's less than or equal to (), keep the same sign.
3. Write two inequalities:
• One where the expression inside the absolute value is positive
• Another where it's negative
4. Solve each inequality separately.
5. Combine the solutions using "and" ().

Example: Solve |x + 2| < 3

Step 1: The absolute value is already isolated.
Step 2: Keep the < sign.
Step 3: Write two inequalities: -3 < x + 2 < 3
Step 4: Solve each part: -5 < x and x < 1
Step 5: Final solution: -5 < x < 1

#### Case 2: Greater Than (>) Inequalities

1. Isolate the absolute value expression on one side of the inequality.
2. If the inequality is strictly greater than (>), keep the inequality sign. If it's greater than or equal to (), keep the same sign.
3. Write two inequalities:
• One where the expression inside the absolute value is greater than the positive right side
• Another where it's less than the negative of the right side
4. Solve each inequality separately.
5. Combine the solutions using "or" ().

Example: Solve |x - 1| > 4

Step 1: The absolute value is already isolated.
Step 2: Keep the > sign.
Step 3: Write two inequalities: x - 1 > 4 or x - 1 < -4
Step 4: Solve each part: x > 5 or x < -3
Step 5: Final solution: x < -3 or x > 5

### Graphical Interpretation

For 'less than' inequalities, the solution represents a range between two points on a number line. For 'greater than' inequalities, the solution represents two separate ranges extending outward from two points.

### Common Mistakes and How to Avoid Them

• Forgetting to consider both positive and negative cases: Always write out both inequalities.
• Using "and" instead of "or" for greater than inequalities: Remember, 'less than' uses "and", while 'greater than' uses "or".
• Incorrect signs when solving inequalities: Be careful when multiplying or dividing by negative numbers, as this flips the inequality sign.
• Misinterpreting the graphical solution: Practice drawing number lines to visualize your solutions.

By following these steps

## Applications of Absolute Value Inequalities

Absolute value inequalities have numerous real-world applications across various fields, including engineering, finance, and quality control. These mathematical tools help solve complex problems and make critical decisions in practical scenarios. Let's explore some concrete examples to understand their significance better.

In engineering, absolute value inequalities are often used to define tolerance ranges for manufacturing processes. For instance, consider a factory producing metal rods for construction. The specification might require the rods to be 10 meters long, with an acceptable deviation of no more than 0.1 meters in either direction. This can be expressed as an absolute value inequality: |x - 10| 0.1, where x represents the actual length of the rod. Any rod satisfying this inequality would be considered within the acceptable range.

The field of finance also utilizes absolute value inequalities, particularly in risk management and investment strategies. For example, a financial advisor might recommend that a client's portfolio should not deviate more than 5% from a target allocation of 60% stocks and 40% bonds. This can be represented as |x - 60| 5, where x is the actual percentage of stocks in the portfolio. Solving this inequality would give the acceptable range of 55% to 65% for the stock allocation.

Quality control in manufacturing processes frequently employs absolute value inequalities to ensure products meet specific standards. Consider a pharmaceutical company producing tablets with a target weight of 500 milligrams. If the acceptable variation is ±10 milligrams, this can be expressed as |x - 500| 10, where x is the actual weight of a tablet. Any tablet weighing between 490 and 510 milligrams would pass the quality check.

Let's look at a word problem to illustrate how to translate a real-world scenario into an absolute value inequality and solve it. Suppose a city's water treatment plant needs to maintain a pH level of 7.0 in its water supply, with a maximum allowable deviation of 0.5 in either direction. How would we express this as an inequality and determine the acceptable pH range?

To solve this, we can set up the absolute value inequality: |x - 7| 0.5, where x represents the actual pH level. To solve this inequality, we consider two cases:

Case 1: x - 7 0.5
x 7.5

Case 2: -(x - 7) 0.5
-x + 7 0.5
-x -6.5
x 6.5

Combining these results, we find that the acceptable pH range is 6.5 x 7.5. This means the water treatment plant must maintain the pH level between 6.5 and 7.5 to meet the required standards.

Another practical application of absolute value inequalities can be found in climate control systems. For instance, a museum might need to maintain a constant temperature of 20°C to preserve delicate artifacts, with a maximum allowable fluctuation of 2°C. This scenario can be represented by the inequality |x - 20| 2, where x is the actual temperature in the museum. Solving this inequality would show that the acceptable temperature range is between 18°C and 22°C.

In conclusion, absolute value inequalities play a crucial role in various real-world applications, from engineering and finance to quality control and environmental management. By translating practical problems into mathematical inequalities, professionals can define acceptable ranges, set tolerances, and make informed decisions based on quantifiable data. Understanding how to formulate and solve these inequalities is essential for anyone working in fields that require precise measurements, risk assessment, or quality assurance.

## Common Mistakes and Troubleshooting

When solving absolute value inequalities, students often encounter several common mistakes that can lead to incorrect solutions. By understanding these errors and implementing effective troubleshooting strategies, students can improve their problem-solving skills and achieve more accurate results.

One frequent mistake is misinterpreting the inequality sign when splitting absolute value equation into two parts. For example, when solving |x - 3| < 5, some students incorrectly write it as x - 3 < 5 and x - 3 > -5. The correct interpretation should be -5 < x - 3 < 5. To avoid this error, always remember that the inequality sign remains the same for both parts when dealing with "less than" (<) inequalities, and it reverses for "greater than" (>) inequalities.

Another common error is forgetting to consider both positive and negative solutions when solving absolute value inequalities. For instance, in |2x + 1| 7, students might only solve for the positive case: 2x + 1 7. However, it's crucial to also consider the negative case: 2x + 1 -7. To prevent this mistake, always solve for both cases and combine the results to obtain the complete solution set.

Students sometimes struggle with graphing absolute value inequalities on a number line. For example, when graphing |x - 2| < 3, they might incorrectly shade the region outside the interval (-1, 5) instead of inside it. To avoid this, carefully consider the inequality sign and remember that the solution represents the values closer to the center point than the given distance.

To troubleshoot and verify solutions, students can employ several strategies. First, always check if the solution satisfies the original inequality by substituting values from the solution set back into the original equation. For instance, if the solution to |x + 1| > 4 is x < -5 or x > 3, test values like -6 and 4 to ensure they satisfy the original inequality.

Another effective strategy is to graph both sides of the inequality and visually compare them. This method can help identify any discrepancies in the solution. Additionally, using online graphing calculators or software can provide a quick way to verify results and catch potential errors.

When correcting incorrect solutions, it's essential to identify the specific error and address it step-by-step. For example, if a student incorrectly solves splitting absolute value equation as 2x - 5 3 and 2x - 5 -3, guide them through the correct process: -3 2x - 5 3, then solve each part separately to obtain 1 x 4 as the final solution.

By being aware of these common mistakes, implementing effective troubleshooting techniques, and practicing error correction, students can significantly improve their ability to solve absolute value inequalities accurately and confidently.

## Conclusion

In this lesson, we've explored the fundamental concepts of absolute value inequalities. We began by reviewing absolute values and their graphical representations on a number line. We then delved into solving both single and compound absolute value inequalities, emphasizing the importance of considering two separate cases for each inequality. Key points covered include interpreting inequality symbols, isolating the absolute value expression, and solving the resulting linear inequalities. Understanding absolute values is crucial for mastering this topic, so we encourage you to revisit the introduction video if needed. To reinforce your learning, practice solving additional practice problems and pay attention to the different inequality types. Remember, proficiency comes with consistent practice. For further engagement, try the practice problems provided in your workbook or explore related videos on our platform. By mastering absolute value inequalities, you'll build a strong foundation for more advanced mathematical concepts. Keep up the great work!

### Example:

Solving Basic Absolute Value Inequalities
Solve: $|x|$ < $4$
$|x| \leq$ $4$

#### Step 1: Understanding the Inequality

The first step in solving the absolute value inequality $|x| \leq 4$ is to understand what the inequality means. The absolute value of a number represents its distance from 0 on the number line. Therefore, $|x| \leq 4$ means that the distance of $x$ from 0 is less than or equal to 4 units.

#### Step 2: Interpreting the Inequality

By definition, the absolute value inequality $|x| \leq 4$ can be interpreted as $x$ being within 4 units of 0. This means that $x$ can be any number between -4 and 4, inclusive. Mathematically, this can be written as: $-4 \leq x \leq 4$

#### Step 3: Graphical Representation

To visualize this inequality, we can draw a number line and mark the points -4 and 4. Since the inequality is less than or equal to, we use closed circles at -4 and 4 to indicate that these points are included in the solution set. The region between -4 and 4, including the endpoints, represents all possible values of $x$ that satisfy the inequality.

#### Step 4: Checking Values

To ensure our interpretation is correct, we can check a few values within and outside the interval. For example:

• Values like 1, -1, 2, and -2 are within the interval and should satisfy the inequality $|x| \leq 4$.
• Values like 5 and -5 are outside the interval and should not satisfy the inequality $|x| \leq 4$.
Indeed, $|1| = 1 \leq 4$, $|-1| = 1 \leq 4$, $|2| = 2 \leq 4$, and $|-2| = 2 \leq 4$. On the other hand, $|5| = 5 \not\leq 4$ and $|-5| = 5 \not\leq 4$.

#### Step 5: Translating to Algebraic Form

Based on our graphical representation and value checks, we can translate the inequality $|x| \leq 4$ into its algebraic form: $-4 \leq x \leq 4$ This means that $x$ can take any value between -4 and 4, inclusive.

#### Step 6: Conclusion

In conclusion, when solving the absolute value inequality $|x| \leq 4$, we find that the solution set includes all values of $x$ that are within 4 units of 0. This can be expressed algebraically as: $-4 \leq x \leq 4$ This solution can be visualized on a number line with closed circles at -4 and 4, indicating that these endpoints are included in the solution set.

### FAQs

1. What is the difference between solving absolute value equations and inequalities?

Absolute value equations typically have a finite number of solutions, while absolute value inequalities often result in a range of solutions. When solving inequalities, you need to consider two cases (positive and negative) and often end up with compound inequalities or interval notation for the solution.

2. How do I know when to use "and" or "or" when solving absolute value inequalities?

Use "and" (intersection) for "less than" (<) inequalities and "or" (union) for "greater than" (>) inequalities. For example, |x| < 5 becomes -5 < x < 5 (using "and"), while |x| > 5 becomes x < -5 or x > 5 (using "or").

3. Can absolute value inequalities have no solution?

Yes, absolute value inequalities can have no solution. For example, |x| < -3 has no solution because the absolute value of a number is always non-negative, so it can never be less than a negative number.

4. How do I graph the solution of an absolute value inequality?

To graph the solution, first solve the inequality algebraically. For "less than" inequalities, the solution will be an interval on the number line. For "greater than" inequalities, the solution will be two separate regions extending outward from two points. Use open circles for strict inequalities and closed circles for inclusive inequalities.

5. What are some real-world applications of absolute value inequalities?

Absolute value inequalities are used in various fields such as engineering (for tolerance ranges), finance (for investment strategies), quality control (for product specifications), and environmental science (for acceptable ranges of measurements). They help define acceptable ranges or limits in many practical scenarios.

### Prerequisite Topics

Understanding the prerequisite topics is crucial for mastering the concept of solving absolute value inequalities. These foundational skills provide the necessary framework to tackle more complex problems with confidence and precision.

One of the most important prerequisites is absolute value functions. Grasping the concept of absolute value is essential, as it forms the core of absolute value inequalities. This knowledge helps students interpret the meaning of absolute value in various mathematical contexts and understand how it affects the solution set.

Another critical skill is solving multi-step linear inequalities. This prerequisite equips students with the ability to manipulate inequalities, which is fundamental when dealing with absolute value inequalities. The techniques learned here, such as isolating variables and maintaining inequality direction, are directly applicable to more complex absolute value problems.

While not directly related, understanding operations on vectors in magnitude and direction form can provide valuable insights into the concept of absolute value, as magnitude is closely related to absolute value in mathematical terms.

Similarly, familiarity with the spread of a data set - standard deviation & variance can enhance a student's understanding of how absolute values relate to distances from a central point, which is often relevant in solving absolute value inequalities.

A crucial stepping stone to solving absolute value inequalities is solving absolute value equations. This prerequisite introduces students to the concept of considering two separate cases when dealing with absolute values, a skill directly transferable to inequalities.

Lastly, the ability to express linear inequalities graphically and algebraically is invaluable. This skill helps students visualize the solution sets of absolute value inequalities on a number line and understand the relationship between algebraic and graphical representations.

By mastering these prerequisite topics, students build a strong foundation for tackling absolute value inequalities. Each concept contributes to a deeper understanding of the subject, allowing for more efficient problem-solving and a more comprehensive grasp of advanced mathematical concepts. Remember, mathematics is a cumulative subject, and a solid understanding of these prerequisites will pave the way for success in solving absolute value inequalities and beyond.

$\bullet$ absolute value inequalities: $| \heartsuit |$ < $a$,
solution:$-a$ < $\heartsuit$ < $a$
$\bullet$ absolute value inequalities: $| \heartsuit |$ > $a$,
solution:$\heartsuit$ < $-a$ $\;or\;$ $\heartsuit$ > $a$