Solving Absolute Value Inequality 3|2x-7|+17≥29 In Interval Notation
In mathematics, inequalities play a crucial role in defining ranges and sets of solutions. When absolute values are introduced into inequalities, the complexity increases, demanding a systematic approach to find accurate solutions. This article delves into the process of solving inequalities involving absolute values and expressing the solutions in interval notation. Using the example inequality 3|2x - 7| + 17 ≥ 29, we will demonstrate a step-by-step method applicable to various problems of this type. Understanding these concepts is essential for students and anyone involved in mathematical analysis, as they form the basis for more advanced topics in calculus and real analysis.
Understanding Absolute Value
Before diving into the solution process, it's essential to understand the concept of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction. Mathematically, the absolute value of a number x, denoted as |x|, is defined as:
|x| = x, if x ≥ 0
|x| = -x, if x < 0
This definition means that the absolute value of a positive number or zero is the number itself, while the absolute value of a negative number is its opposite (which is positive). For instance, |5| = 5 and |-5| = 5. The absolute value function ensures that the result is always non-negative.
When solving inequalities involving absolute values, we must consider two cases: one where the expression inside the absolute value is positive or zero and another where it is negative. This approach stems directly from the definition of absolute value and ensures that we account for all possible scenarios. Understanding this dual nature is the key to correctly solving these types of inequalities. In the context of our example, 3|2x - 7| + 17 ≥ 29, we will eventually need to consider both when 2x - 7 is positive or zero and when it is negative to find the complete solution set.
Step 1: Isolate the Absolute Value Expression
The first step in solving an absolute value inequality is to isolate the absolute value expression on one side of the inequality. This means performing algebraic operations to get the term containing the absolute value by itself. In our example, the inequality is 3|2x - 7| + 17 ≥ 29. To isolate the absolute value expression, |2x - 7|, we first need to subtract 17 from both sides of the inequality:
3|2x - 7| + 17 - 17 ≥ 29 - 17
This simplifies to:
3|2x - 7| ≥ 12
Next, we divide both sides by 3 to completely isolate the absolute value:
(3|2x - 7|)/3 ≥ 12/3
Which gives us:
|2x - 7| ≥ 4
Now that the absolute value expression is isolated, we can proceed to the next step, which involves considering the two separate cases based on the definition of absolute value. This isolation step is crucial because it sets the stage for applying the absolute value definition properly. Without isolating the absolute value, it's difficult to determine the correct approach for solving the inequality. The result, |2x - 7| ≥ 4, tells us that the distance of 2x - 7 from zero is greater than or equal to 4.
Step 2: Split the Inequality into Two Cases
Once the absolute value expression is isolated, the next critical step is to split the inequality into two separate cases. This is because the absolute value function has two possible outcomes for any input: the input itself if it is non-negative, and the negation of the input if it is negative. For the inequality |2x - 7| ≥ 4, we consider two cases:
Case 1: The expression inside the absolute value is non-negative.
In this case, 2x - 7 is greater than or equal to zero, so the absolute value |2x - 7| is simply 2x - 7. Thus, we have:
2x - 7 ≥ 4
This inequality represents the scenario where 2x - 7 is a positive number or zero, and its distance from zero is greater than or equal to 4. We will solve this inequality to find the values of x that satisfy this condition.
Case 2: The expression inside the absolute value is negative.
In this case, 2x - 7 is less than zero, so the absolute value |2x - 7| is the negation of 2x - 7, which is -(2x - 7). Thus, we have:
-(2x - 7) ≥ 4
This inequality represents the scenario where 2x - 7 is a negative number, and its distance from zero (which is its negation) is greater than or equal to 4. Again, we will solve this inequality to find the values of x that satisfy this condition. Splitting the absolute value inequality into these two cases is essential because it covers all possibilities and ensures that we find the complete solution set. Each case will give us a different set of solutions, and we will combine these sets to get the final solution.
Step 3: Solve Each Inequality Separately
After splitting the absolute value inequality into two cases, the next step is to solve each inequality separately. This involves applying standard algebraic techniques to isolate the variable x in each case. Let's solve the inequalities we derived in the previous step.
Solving Case 1: 2x - 7 ≥ 4
To solve this inequality, we first add 7 to both sides:
2x - 7 + 7 ≥ 4 + 7
This simplifies to:
2x ≥ 11
Next, we divide both sides by 2:
(2x)/2 ≥ 11/2
Which gives us:
x ≥ 11/2
So, the solution for Case 1 is all x values that are greater than or equal to 11/2. This means that any number from 11/2 (or 5.5) to positive infinity will satisfy the original absolute value inequality in the context of Case 1.
Solving Case 2: -(2x - 7) ≥ 4
To solve this inequality, we first distribute the negative sign:
-2x + 7 ≥ 4
Next, we subtract 7 from both sides:
-2x + 7 - 7 ≥ 4 - 7
This simplifies to:
-2x ≥ -3
Now, we divide both sides by -2. Remember that when dividing or multiplying an inequality by a negative number, we must reverse the direction of the inequality sign:
(-2x)/(-2) ≤ (-3)/(-2)
Which gives us:
x ≤ 3/2
So, the solution for Case 2 is all x values that are less than or equal to 3/2. This means that any number from negative infinity up to 3/2 (or 1.5) will satisfy the original absolute value inequality in the context of Case 2. Solving each inequality separately allows us to find the ranges of x values that satisfy the original condition under different scenarios, which is crucial for obtaining the complete solution.
Step 4: Express the Solution in Interval Notation
After solving each inequality separately, the final step is to combine the solutions and express them in interval notation. Interval notation is a way of writing sets of numbers using intervals, which are defined by their endpoints and whether the endpoints are included in the set. The symbols used in interval notation are brackets [] for included endpoints and parentheses () for excluded endpoints. The infinity symbols ∞ and -∞ are used to represent unbounded intervals. From our previous steps, we have two solution sets:
- x ≥ 11/2
- x ≤ 3/2
Let's express each of these in interval notation:
Solution Set 1: x ≥ 11/2
This inequality represents all numbers greater than or equal to 11/2. In interval notation, this is written as:
[11/2, ∞)
The square bracket [ indicates that 11/2 is included in the interval, and the parenthesis ) indicates that infinity is not included (since infinity is not a number, we cannot include it in a set).
Solution Set 2: x ≤ 3/2
This inequality represents all numbers less than or equal to 3/2. In interval notation, this is written as:
(-∞, 3/2]
The parenthesis ( indicates that negative infinity is not included, and the square bracket ] indicates that 3/2 is included in the interval.
Combining the Solution Sets
Since the original absolute value inequality is satisfied if either Case 1 or Case 2 is true, we need to combine the two solution sets. This is done by taking the union of the two intervals. The union of two sets includes all elements that are in either set. In interval notation, the union is represented by the symbol ∪. Therefore, the complete solution to the inequality |2x - 7| ≥ 4 is:
(-∞, 3/2] ∪ [11/2, ∞)
This interval notation represents all numbers that are either less than or equal to 3/2 or greater than or equal to 11/2. This comprehensive solution set fully captures the range of values that satisfy the original absolute value inequality.
Conclusion
Solving inequalities involving absolute values requires a systematic approach that includes isolating the absolute value expression, splitting the inequality into two cases, solving each case separately, and expressing the solution in interval notation. By following these steps, we can accurately determine the solution set for a wide range of absolute value inequalities. The example 3|2x - 7| + 17 ≥ 29 illustrates this process effectively. Understanding these techniques is crucial for various mathematical applications, making it an essential skill for students and professionals alike. The use of interval notation provides a clear and concise way to represent solution sets, further enhancing our ability to communicate mathematical results accurately.
By mastering the methods outlined in this article, readers will be well-equipped to tackle absolute value inequalities and express their solutions effectively, paving the way for success in more advanced mathematical studies.
