Solving Absolute Value Inequalities A Step By Step Guide
Absolute value inequalities can seem daunting at first, but with a systematic approach and a clear understanding of the properties of absolute values, they can be solved effectively. This article serves as a comprehensive guide to tackling such inequalities, specifically focusing on the solutions to four challenging problems |x - 4| < 2|x - 5|, |3 - x| > 3|x|, |x - 2| ≥ 4|x + 2|, and |5x + 4| < 2|x - 1|. We will break down each problem step-by-step, providing clear explanations and strategies to help you master this crucial mathematical concept.
Understanding Absolute Value
Before diving into the specific problems, it's essential to grasp the fundamental concept of absolute value. The absolute value of a number represents its distance from zero on the number line. It 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
This definition is the cornerstone for solving absolute value inequalities. When dealing with inequalities involving absolute values, we need to consider both cases: the expression inside the absolute value is positive or zero, and the expression inside the absolute value is negative. This often leads to splitting the problem into different intervals, solving the inequality within each interval, and then combining the solutions.
To effectively solve these inequalities, remember that understanding the absolute value definition is paramount. Think of |x| as the distance from x to 0. This distance concept helps in visualizing the solutions on a number line, which can be particularly useful when dealing with inequalities. Also, remember that when you multiply or divide an inequality by a negative number, you must reverse the inequality sign. This is a common pitfall in solving inequalities, so pay close attention to the signs throughout your calculations. Furthermore, when dealing with inequalities involving multiple absolute values, it's crucial to identify the critical points where the expressions inside the absolute values change signs. These critical points will help you divide the number line into intervals and solve the inequality separately within each interval. This piecewise approach ensures that you consider all possible cases and arrive at the correct solution.
(a) Solving |x - 4| < 2|x - 5|
To solve the inequality |x - 4| < 2|x - 5|, we need to consider the critical points where the expressions inside the absolute values change signs. These points are x = 4 and x = 5. This divides the number line into three intervals: x < 4, 4 ≤ x < 5, and x ≥ 5. We will solve the inequality separately in each interval.
Interval 1: x < 4
In this interval, both (x - 4) and (x - 5) are negative. Therefore, |x - 4| = -(x - 4) = 4 - x and |x - 5| = -(x - 5) = 5 - x. Substituting these into the inequality, we get:
4 - x < 2(5 - x) 4 - x < 10 - 2x x < 6
Since we are considering the interval x < 4, the solution in this interval is x < 4 (as all values less than 4 are also less than 6).
Interval 2: 4 ≤ x < 5
In this interval, (x - 4) is non-negative, and (x - 5) is negative. Therefore, |x - 4| = x - 4 and |x - 5| = -(x - 5) = 5 - x. Substituting these into the inequality, we get:
x - 4 < 2(5 - x) x - 4 < 10 - 2x 3x < 14 x < 14/3
Since we are considering the interval 4 ≤ x < 5, we need to find the intersection of 4 ≤ x < 5 and x < 14/3. Since 14/3 ≈ 4.67, the solution in this interval is 4 ≤ x < 14/3.
Interval 3: x ≥ 5
In this interval, both (x - 4) and (x - 5) are non-negative. Therefore, |x - 4| = x - 4 and |x - 5| = x - 5. Substituting these into the inequality, we get:
x - 4 < 2(x - 5) x - 4 < 2x - 10 x > 6
Since we are considering the interval x ≥ 5, the solution in this interval is x > 6.
Combining the Solutions
Combining the solutions from all three intervals, we get x < 14/3 or x > 6. In interval notation, the solution is (-∞, 14/3) ∪ (6, ∞).
When tackling these absolute value inequalities, a strategic approach is essential. Identifying the critical points is the first key step. These are the points where the expressions inside the absolute value symbols equal zero. These points divide the number line into intervals, and within each interval, the signs of the expressions inside the absolute values remain constant. This allows you to rewrite the absolute value expressions without the absolute value signs, simplifying the inequality. Remember to consider the endpoints of these intervals carefully, as they may or may not be included in the solution depending on the inequality sign (strict or non-strict). Solving the inequality within each interval is a standard algebraic process, but always remember to check your solutions against the interval you're working in. The final step is crucial: combine the solutions from all intervals to get the overall solution set. This might involve taking the union or intersection of intervals, so a clear understanding of set operations is beneficial. Visualizing the solution on a number line can be incredibly helpful in this process, especially when dealing with multiple intervals.
(b) Solving |3 - x| > 3|x|
To solve the inequality |3 - x| > 3|x|, we again identify the critical points: x = 0 and x = 3. This divides the number line into three intervals: x < 0, 0 ≤ x < 3, and x ≥ 3.
Interval 1: x < 0
In this interval, (3 - x) is positive, and x is negative. Therefore, |3 - x| = 3 - x and |x| = -x. Substituting these into the inequality, we get:
3 - x > 3(-x) 3 - x > -3x 2x > -3 x > -3/2
Since we are considering the interval x < 0, the solution in this interval is -3/2 < x < 0.
Interval 2: 0 ≤ x < 3
In this interval, (3 - x) is non-negative, and x is non-negative. Therefore, |3 - x| = 3 - x and |x| = x. Substituting these into the inequality, we get:
3 - x > 3x 3 > 4x x < 3/4
Since we are considering the interval 0 ≤ x < 3, the solution in this interval is 0 ≤ x < 3/4.
Interval 3: x ≥ 3
In this interval, (3 - x) is negative, and x is non-negative. Therefore, |3 - x| = -(3 - x) = x - 3 and |x| = x. Substituting these into the inequality, we get:
x - 3 > 3x -2x > 3 x < -3/2
There is no solution in this interval since we are considering x ≥ 3, and the solution x < -3/2 is contradictory.
Combining the Solutions
Combining the solutions from the first two intervals, we get -3/2 < x < 3/4. In interval notation, the solution is (-3/2, 3/4).
When dealing with inequalities like |3 - x| > 3|x|, recognizing the critical points is vitally important. These points, where the expressions inside the absolute values become zero, act as boundaries that define different intervals on the number line. Within each interval, the absolute value expressions can be replaced with their equivalent algebraic expressions, making the inequality easier to solve. Remember that the absolute value of an expression changes its sign depending on whether the expression is positive or negative. By identifying the intervals correctly, you ensure that you're using the appropriate algebraic form of the absolute value in each case. This piecewise approach is a fundamental technique for solving any inequality involving absolute values. After solving the inequality within each interval, it's crucial to combine the solutions carefully, paying attention to the interval boundaries and whether they should be included or excluded from the final solution. A number line representation can be immensely helpful in visualizing the intervals and their corresponding solutions, making the process of combining solutions more intuitive and less prone to errors. Always double-check your final solution to ensure it satisfies the original inequality.
(c) Solving |x - 2| ≥ 4|x + 2|
For the inequality |x - 2| ≥ 4|x + 2|, the critical points are x = 2 and x = -2. This gives us three intervals: x < -2, -2 ≤ x < 2, and x ≥ 2.
Interval 1: x < -2
In this interval, both (x - 2) and (x + 2) are negative. Therefore, |x - 2| = -(x - 2) = 2 - x and |x + 2| = -(x + 2) = -x - 2. Substituting these into the inequality, we get:
2 - x ≥ 4(-x - 2) 2 - x ≥ -4x - 8 3x ≥ -10 x ≥ -10/3
Since we are considering the interval x < -2, the solution in this interval is -10/3 ≤ x < -2.
Interval 2: -2 ≤ x < 2
In this interval, (x - 2) is negative, and (x + 2) is non-negative. Therefore, |x - 2| = -(x - 2) = 2 - x and |x + 2| = x + 2. Substituting these into the inequality, we get:
2 - x ≥ 4(x + 2) 2 - x ≥ 4x + 8 -5x ≥ 6 x ≤ -6/5
Since we are considering the interval -2 ≤ x < 2, the solution in this interval is -2 ≤ x ≤ -6/5.
Interval 3: x ≥ 2
In this interval, both (x - 2) and (x + 2) are non-negative. Therefore, |x - 2| = x - 2 and |x + 2| = x + 2. Substituting these into the inequality, we get:
x - 2 ≥ 4(x + 2) x - 2 ≥ 4x + 8 -3x ≥ 10 x ≤ -10/3
There is no solution in this interval since we are considering x ≥ 2, and the solution x ≤ -10/3 is contradictory.
Combining the Solutions
Combining the solutions from the first two intervals, we get -10/3 ≤ x ≤ -6/5. In interval notation, the solution is [-10/3, -6/5].
The approach to solving the inequality |x - 2| ≥ 4|x + 2| involves the same fundamental principle: identifying critical points and analyzing intervals. However, the key here is to ensure meticulous algebraic manipulation and careful consideration of the inequality sign. Remember that when you multiply or divide an inequality by a negative number, you must reverse the inequality sign. This is a common source of errors, so double-check each step involving multiplication or division by a negative coefficient. Another important aspect is the interpretation of the inequality itself. The inequality |x - 2| ≥ 4|x + 2| implies that the distance between x and 2 is greater than or equal to four times the distance between x and -2. This geometric interpretation can sometimes provide a valuable check on your algebraic solution. If your algebraic solution doesn't align with this geometric intuition, it might indicate a potential error in your calculations. Always strive to connect the algebraic steps with the underlying geometric meaning to gain a deeper understanding and build confidence in your solution.
(d) Solving |5x + 4| < 2|x - 1|
Finally, let's solve the inequality |5x + 4| < 2|x - 1|. The critical points are x = -4/5 and x = 1. This divides the number line into three intervals: x < -4/5, -4/5 ≤ x < 1, and x ≥ 1.
Interval 1: x < -4/5
In this interval, both (5x + 4) and (x - 1) are negative. Therefore, |5x + 4| = -(5x + 4) = -5x - 4 and |x - 1| = -(x - 1) = 1 - x. Substituting these into the inequality, we get:
-5x - 4 < 2(1 - x) -5x - 4 < 2 - 2x -3x < 6 x > -2
Since we are considering the interval x < -4/5, the solution in this interval is -2 < x < -4/5.
Interval 2: -4/5 ≤ x < 1
In this interval, (5x + 4) is non-negative, and (x - 1) is negative. Therefore, |5x + 4| = 5x + 4 and |x - 1| = -(x - 1) = 1 - x. Substituting these into the inequality, we get:
5x + 4 < 2(1 - x) 5x + 4 < 2 - 2x 7x < -2 x < -2/7
Since we are considering the interval -4/5 ≤ x < 1, the solution in this interval is -4/5 ≤ x < -2/7.
Interval 3: x ≥ 1
In this interval, both (5x + 4) and (x - 1) are non-negative. Therefore, |5x + 4| = 5x + 4 and |x - 1| = x - 1. Substituting these into the inequality, we get:
5x + 4 < 2(x - 1) 5x + 4 < 2x - 2 3x < -6 x < -2
There is no solution in this interval since we are considering x ≥ 1, and the solution x < -2 is contradictory.
Combining the Solutions
Combining the solutions from the first two intervals, we get -2 < x < -2/7. In interval notation, the solution is (-2, -2/7).
For inequalities like |5x + 4| < 2|x - 1|, the algebraic manipulations can become more intricate, increasing the likelihood of making a mistake. Therefore, meticulousness and careful attention to detail are paramount. After obtaining the solutions within each interval, it's a good practice to test a value within that interval in the original inequality to confirm that it indeed satisfies the condition. This provides a valuable check on your calculations and helps catch any potential errors. Furthermore, consider the graphical interpretation of the inequality. The inequality |5x + 4| < 2|x - 1| can be interpreted as the graph of y = |5x + 4| lying below the graph of y = 2|x - 1|. Sketching these graphs can provide a visual representation of the solution set and help you verify your algebraic solution. This multi-faceted approach, combining algebraic techniques with numerical and graphical checks, is a powerful way to build confidence in your solution and develop a deeper understanding of absolute value inequalities.
Conclusion
Solving absolute value inequalities requires a systematic approach involving identifying critical points, dividing the number line into intervals, solving the inequality in each interval, and combining the solutions. By understanding the definition of absolute value and practicing these techniques, you can confidently tackle a wide range of absolute value inequality problems. Remember to always check your solutions and consider the geometric interpretation of the inequality to ensure accuracy and build a deeper understanding of the concepts.
