Solving Absolute Value Inequality |4x-15|+5 ≤ 20 A Step-by-Step Guide
Understanding and solving absolute value inequalities is a fundamental skill in mathematics. These inequalities, which involve expressions within absolute value symbols, often appear in various mathematical contexts, from algebra to calculus. In this comprehensive guide, we will delve into the intricacies of solving the absolute value inequality |4x - 15| + 5 ≤ 20, providing a step-by-step approach that not only solves the problem but also enhances your understanding of the underlying principles. By mastering these techniques, you will be well-equipped to tackle similar problems with confidence and precision.
Understanding Absolute Value
Before we dive into the solution, it's crucial to understand the concept of absolute value. The absolute value of a number is its distance from zero on the number line. This distance is always non-negative. Mathematically, the absolute value of a number x, denoted as |x|, is defined as:
|x| = x, if x ≥ 0
|x| = -x, if x < 0
For instance, |5| = 5 and |-5| = 5. The absolute value essentially disregards the sign of the number, focusing solely on its magnitude. This concept is critical when dealing with inequalities, as it introduces two possible scenarios that must be considered.
When solving absolute value inequalities, we leverage this definition to split the problem into two separate cases. This is because the expression inside the absolute value can be either positive or negative, and both possibilities need to be accounted for to find the complete solution set. Failing to consider both cases can lead to an incomplete or incorrect answer. The process involves isolating the absolute value expression and then setting up two inequalities that represent the two scenarios, thereby capturing all possible solutions.
Step-by-Step Solution of |4x - 15| + 5 ≤ 20
Now, let's tackle the given inequality |4x - 15| + 5 ≤ 20 step by step. This process will illustrate how to apply the principles of absolute value to solve a specific problem, offering a practical understanding that can be generalized to other similar inequalities. Each step is designed to build upon the previous one, ensuring clarity and accuracy in the solution process. By the end of this section, you will have a thorough understanding of how to approach and solve absolute value inequalities effectively.
Step 1: Isolate the Absolute Value Expression
Our first goal is to isolate the absolute value expression, |4x - 15|, on one side of the inequality. To achieve this, we need to eliminate the '+ 5' term on the left side. We can do this by subtracting 5 from both sides of the inequality. This maintains the balance of the inequality and brings us closer to isolating the absolute value term.
Original inequality: |4x - 15| + 5 ≤ 20
Subtract 5 from both sides: |4x - 15| + 5 - 5 ≤ 20 - 5
Simplified inequality: |4x - 15| ≤ 15
Now that we have isolated the absolute value expression, we can proceed to the next step, which involves splitting the inequality into two separate cases. This is a crucial step in solving absolute value inequalities, as it allows us to account for both positive and negative possibilities of the expression inside the absolute value.
Step 2: Split the Inequality into Two Cases
The definition of absolute value necessitates that we consider two cases: the expression inside the absolute value is either non-negative or negative. This is because the absolute value of a number is its distance from zero, and distance is always non-negative. Therefore, we must account for both possibilities to find the complete solution set.
Case 1: The expression inside the absolute value is non-negative (4x - 15 ≥ 0)
In this case, the absolute value does not change the sign, so we have:
4x - 15 ≤ 15
Case 2: The expression inside the absolute value is negative (4x - 15 < 0)
In this case, the absolute value changes the sign, so we have:
-(4x - 15) ≤ 15
These two cases represent the core of solving absolute value inequalities. By addressing each case separately, we can determine the range of values for x that satisfy the original inequality. The next step involves solving each of these inequalities individually.
Step 3: Solve Each Inequality
Now that we have split the absolute value inequality into two cases, we need to solve each case separately to find the potential values of x. This involves applying algebraic techniques to isolate x in each inequality. The solutions from both cases will then be combined to form the complete solution set for the original inequality.
Solving Case 1: 4x - 15 ≤ 15
To solve this inequality, we will first add 15 to both sides:
4x - 15 + 15 ≤ 15 + 15
4x ≤ 30
Next, we divide both sides by 4:
4x / 4 ≤ 30 / 4
x ≤ 7.5
So, the solution for Case 1 is x ≤ 7.5. This means that any value of x less than or equal to 7.5 satisfies the inequality in the first case.
Solving Case 2: -(4x - 15) ≤ 15
First, we distribute the negative sign:
-4x + 15 ≤ 15
Next, we subtract 15 from both sides:
-4x + 15 - 15 ≤ 15 - 15
-4x ≤ 0
Now, we divide both sides by -4. Remember that when we divide or multiply an inequality by a negative number, we must reverse the inequality sign:
-4x / -4 ≥ 0 / -4
x ≥ 0
So, the solution for Case 2 is x ≥ 0. This means that any value of x greater than or equal to 0 satisfies the inequality in the second case.
By solving both cases, we have identified two ranges of values for x that potentially satisfy the original inequality. The next step is to combine these solutions to find the complete solution set.
Step 4: Combine the Solutions
After solving each case of the absolute value inequality, we need to combine the solutions to find the complete solution set for the original inequality. This involves considering the ranges of values for x that satisfy either case and expressing the final solution in a clear and concise manner. The solution set represents all the values of x that make the original inequality true.
From Case 1, we found that x ≤ 7.5. This means that all values of x less than or equal to 7.5 satisfy the inequality |4x - 15| ≤ 15 under the condition that 4x - 15 is non-negative.
From Case 2, we found that x ≥ 0. This means that all values of x greater than or equal to 0 satisfy the inequality |4x - 15| ≤ 15 under the condition that 4x - 15 is negative.
To combine these solutions, we consider the intersection of these two ranges. In other words, we are looking for the values of x that satisfy both x ≤ 7.5 and x ≥ 0. This can be expressed as a compound inequality:
0 ≤ x ≤ 7.5
This compound inequality represents the complete solution set for the original absolute value inequality. It indicates that any value of x between 0 and 7.5, inclusive, will satisfy the inequality |4x - 15| + 5 ≤ 20.
Step 5: Express the Solution
The final step in solving the absolute value inequality is to express the solution in a clear and standard format. This ensures that the solution is easily understood and can be readily applied in other contexts. The solution can be expressed in several ways, including inequality notation, interval notation, and graphically on a number line.
Inequality Notation:
As we determined in the previous step, the solution to the inequality |4x - 15| + 5 ≤ 20 is:
0 ≤ x ≤ 7.5
This notation clearly shows the range of values that x can take. It indicates that x is greater than or equal to 0 and less than or equal to 7.5.
Interval Notation:
Interval notation is another common way to express the solution set. It uses brackets and parentheses to indicate whether the endpoints are included in the solution. A square bracket [ or ] indicates that the endpoint is included, while a parenthesis ( or ) indicates that the endpoint is not included.
In this case, since x can be equal to both 0 and 7.5, we use square brackets:
[0, 7.5]
This interval notation represents the same solution set as the inequality notation, but in a more compact form.
Graphical Representation:
The solution can also be represented graphically on a number line. This provides a visual understanding of the range of values that satisfy the inequality. To represent the solution on a number line, we draw a line and mark the endpoints 0 and 7.5. Since the endpoints are included in the solution, we use closed circles (or brackets) at these points. Then, we shade the region between 0 and 7.5 to indicate that all values in this range are part of the solution.
The graphical representation provides an intuitive understanding of the solution set, making it clear which values of x satisfy the inequality.
By expressing the solution in these various formats, we ensure that it is easily understood and can be used effectively in different mathematical contexts.
Conclusion
In this comprehensive guide, we have explored the process of solving the absolute value inequality |4x - 15| + 5 ≤ 20. We began by understanding the fundamental concept of absolute value and its implications for solving inequalities. We then proceeded step by step, isolating the absolute value expression, splitting the inequality into two cases, solving each case individually, and combining the solutions. Finally, we expressed the solution in various formats, including inequality notation, interval notation, and graphical representation.
The key takeaways from this guide include:
- Understanding Absolute Value: The absolute value of a number is its distance from zero, which is always non-negative.
- Isolating the Absolute Value Expression: This is the first step in solving any absolute value inequality.
- Splitting into Two Cases: This is necessary to account for both positive and negative possibilities of the expression inside the absolute value.
- Solving Each Case: Apply algebraic techniques to isolate the variable in each inequality.
- Combining Solutions: Consider the intersection of the solution sets from each case.
- Expressing the Solution: Use appropriate notation (inequality, interval, or graphical) to clearly represent the solution set.
By mastering these steps, you will be well-equipped to solve a wide range of absolute value inequalities. Remember to practice and apply these techniques to reinforce your understanding and build confidence in your problem-solving abilities. Absolute value inequalities are a crucial topic in mathematics, and a solid grasp of these concepts will benefit you in various mathematical endeavors.
This guide has provided a thorough explanation and a step-by-step solution to the given inequality. By understanding the underlying principles and practicing these techniques, you can confidently tackle similar problems and further enhance your mathematical skills.
