Solve Absolute Value Inequalities & Graph Solutions
Hey guys! Today, we're diving into the exciting world of solving inequalities involving absolute values and visualizing these solutions on the number line. This is a crucial concept in mathematics, forming the backbone for more advanced topics in algebra and calculus. We'll break down the process step-by-step, ensuring you grasp the fundamental principles and can confidently tackle any problem that comes your way. So, let's get started and unravel the mysteries of absolute value inequalities!
Understanding Absolute Value
Before we jump into solving inequalities, let's quickly recap what absolute value means. In simple terms, the absolute value of a number is its distance from zero, regardless of direction. This means that the absolute value is always non-negative. For example, the absolute value of 5, written as |5|, is 5 because 5 is 5 units away from zero. Similarly, the absolute value of -5, written as |-5|, is also 5 because -5 is also 5 units away from zero. Understanding this fundamental concept is key to solving inequalities involving absolute values. Remember, absolute value deals with distance, and distance is always a positive quantity or zero.
Now, let's consider how this translates to inequalities. When we see an inequality like |x| ≤ 5, it means we're looking for all the numbers x whose distance from zero is less than or equal to 5. This includes numbers like 0, 1, -1, 2, -2, and so on, up to 5 and -5. Conversely, if we have |x| ≥ 5, we're looking for numbers whose distance from zero is greater than or equal to 5. This would include numbers like 5, -5, 6, -6, 7, -7, and so on. The key takeaway here is that absolute value inequalities often lead to two separate inequalities that we need to solve. Mastering this interpretation is crucial for successfully navigating these types of problems. We'll see how this plays out in detail as we work through specific examples.
Solving |x| ≤ 5
Let's start with our first inequality: |x| ≤ 5. To solve this, we need to consider what it means for the absolute value of x to be less than or equal to 5. This means that x can be any number whose distance from zero is 5 or less. This gives us two separate inequalities to consider: x ≤ 5 and x ≥ -5. Think about it this way: if a number is within 5 units of zero, it must be between -5 and 5, inclusive. This split into two inequalities is a fundamental step in solving absolute value inequalities.
So, we have -5 ≤ x ≤ 5. This represents all the numbers between -5 and 5, including -5 and 5 themselves. To represent this solution on the number line, we draw a number line and mark the points -5 and 5. Since the inequality includes “equal to,” we use closed circles (or brackets) at -5 and 5 to indicate that these points are part of the solution. Then, we shade the region between -5 and 5, indicating that all the numbers in this interval are solutions. Visualizing the solution on the number line makes it much easier to understand the range of values that satisfy the inequality. The shaded region clearly shows all the numbers that are within 5 units of zero.
Solving |x + 6/2| ≥ 12
Now, let's tackle the inequality |x + 6/2| ≥ 12. First, we can simplify the expression inside the absolute value: 6/2 = 3, so the inequality becomes |x + 3| ≥ 12. This inequality tells us that the distance between x + 3 and zero is greater than or equal to 12. Just like before, this leads to two separate inequalities. We have x + 3 ≥ 12 or x + 3 ≤ -12. Remember, the absolute value creates two possibilities: the expression inside is either greater than or equal to the positive value, or it's less than or equal to the negative value.
To solve the first inequality, x + 3 ≥ 12, we subtract 3 from both sides, giving us x ≥ 9. This means any number greater than or equal to 9 is a solution. For the second inequality, x + 3 ≤ -12, we also subtract 3 from both sides, resulting in x ≤ -15. This means any number less than or equal to -15 is also a solution. So, our solution set consists of two separate intervals: x ≥ 9 and x ≤ -15. Solving each inequality separately is key to finding the complete solution set.
To represent this solution on the number line, we mark the points 9 and -15. Since the inequalities include “equal to,” we use closed circles (or brackets) at 9 and -15. Then, we shade the region to the right of 9, representing x ≥ 9, and the region to the left of -15, representing x ≤ -15. This gives us two separate shaded regions, indicating that the solutions are all numbers that are either greater than or equal to 9 or less than or equal to -15. The number line visualization clearly shows the two distinct intervals that satisfy the inequality.
Solving |3x + 5| ≥ |4 + x|
Finally, let's tackle the inequality |3x + 5| ≥ |4 + x|. This one looks a bit more complex because we have absolute values on both sides. The key to solving these types of inequalities is to consider the different cases that arise from the possible signs of the expressions inside the absolute values. We have four possible cases:
- Both expressions are positive or zero: 3x + 5 ≥ 0 and 4 + x ≥ 0. In this case, we can simply remove the absolute value signs and solve the inequality 3x + 5 ≥ 4 + x.
- Both expressions are negative: 3x + 5 < 0 and 4 + x < 0. In this case, we remove the absolute value signs and multiply both sides of the inequality by -1, which reverses the inequality sign: -(3x + 5) ≥ -(4 + x).
- 3x + 5 is positive or zero, and 4 + x is negative: 3x + 5 ≥ 0 and 4 + x < 0. Here, we have 3x + 5 ≥ -(4 + x).
- 3x + 5 is negative, and 4 + x is positive or zero: 3x + 5 < 0 and 4 + x ≥ 0. Here, we have -(3x + 5) ≥ 4 + x.
Breaking the problem into cases is essential when dealing with absolute values on both sides of an inequality. Let's solve each case individually.
Case 1: 3x + 5 ≥ 0 and 4 + x ≥ 0
First, let's solve the conditions 3x + 5 ≥ 0 and 4 + x ≥ 0. For 3x + 5 ≥ 0, we subtract 5 from both sides and divide by 3, giving us x ≥ -5/3. For 4 + x ≥ 0, we subtract 4 from both sides, giving us x ≥ -4. Now, we need to solve the inequality 3x + 5 ≥ 4 + x. Subtracting x from both sides gives 2x + 5 ≥ 4. Subtracting 5 from both sides gives 2x ≥ -1. Dividing by 2 gives x ≥ -1/2. So, in this case, we have three conditions: x ≥ -5/3, x ≥ -4, and x ≥ -1/2. The most restrictive condition is x ≥ -1/2, so this is the solution for this case. Identifying the most restrictive condition ensures we capture the correct solutions within the given constraints.
Case 2: 3x + 5 < 0 and 4 + x < 0
Now, let's consider the case where both expressions are negative: 3x + 5 < 0 and 4 + x < 0. This gives us x < -5/3 and x < -4. The more restrictive condition here is x < -4. Next, we solve the inequality -(3x + 5) ≥ -(4 + x). Multiplying both sides by -1 and reversing the inequality sign gives 3x + 5 ≤ 4 + x. Subtracting x from both sides gives 2x + 5 ≤ 4. Subtracting 5 from both sides gives 2x ≤ -1. Dividing by 2 gives x ≤ -1/2. So, in this case, we have the conditions x < -4 and x ≤ -1/2. The solution for this case is x < -4. Remember to reverse the inequality sign when multiplying or dividing by a negative number.
Case 3: 3x + 5 ≥ 0 and 4 + x < 0
In this case, we have 3x + 5 ≥ 0 and 4 + x < 0, which means x ≥ -5/3 and x < -4. These conditions cannot be satisfied simultaneously, as there is no number that is both greater than or equal to -5/3 and less than -4. Therefore, there is no solution in this case. Checking for contradictory conditions is an important step in the process.
Case 4: 3x + 5 < 0 and 4 + x ≥ 0
Finally, let's consider the case where 3x + 5 < 0 and 4 + x ≥ 0, which means x < -5/3 and x ≥ -4. The inequality we need to solve is -(3x + 5) ≥ 4 + x. Expanding and simplifying gives -3x - 5 ≥ 4 + x. Adding 3x to both sides gives -5 ≥ 4 + 4x. Subtracting 4 from both sides gives -9 ≥ 4x. Dividing by 4 gives x ≤ -9/4. So, in this case, we have the conditions x < -5/3, x ≥ -4, and x ≤ -9/4. Converting to decimals, we have x < -1.67, x ≥ -4, and x ≤ -2.25. Combining these conditions, the solution for this case is -4 ≤ x ≤ -9/4. Careful conversion to decimals can help in comparing and combining the inequalities.
Combining the Solutions
Now that we've solved each case, we need to combine the solutions to get the complete solution set for the original inequality. We found the following solutions:
- Case 1: x ≥ -1/2
- Case 2: x < -4
- Case 3: No solution
- Case 4: -4 ≤ x ≤ -9/4
Combining these solutions, we get x < -4 or -4 ≤ x ≤ -9/4 or x ≥ -1/2. This simplifies to x ≤ -9/4 or x ≥ -1/2. Combining the solutions from each case gives us the complete picture.
To represent this on the number line, we mark the points -9/4 and -1/2. We use a closed circle (or bracket) at -9/4 and a closed circle (or bracket) at -1/2. Then, we shade the region to the left of -9/4 and the region to the right of -1/2. This visually represents all the numbers that satisfy the original inequality. The number line provides a clear visualization of the final solution set.
Conclusion
Solving inequalities involving absolute values requires a careful and systematic approach. We need to consider the definition of absolute value, break the problem into cases, solve each case separately, and then combine the solutions. Representing the solutions on the number line provides a valuable visual aid for understanding the solution set. By mastering these techniques, you'll be well-equipped to tackle any absolute value inequality that comes your way. Keep practicing, and you'll become a pro in no time! Remember, guys, practice makes perfect, so keep those pencils moving and those brains buzzing!
