Which Absolute Value Inequalities Have All Real Numbers As Their Solution?

Absolute value inequalities can sometimes seem tricky, but understanding their fundamental properties can make solving them much easier. In particular, we're going to dive into absolute value inequalities that hold true for all real numbers. This happens when the inequality is structured in a way that the absolute value expression always satisfies the condition, regardless of the value of the variable. To identify these types of inequalities, it's crucial to recognize how absolute values behave and how they interact with inequalities.

The absolute value of a number represents its distance from zero on the number line. This means the absolute value is always non-negative (either zero or positive). When we're dealing with inequalities, this non-negativity is key. For instance, if we have an inequality stating that an absolute value expression is greater than a negative number, it will always be true because absolute values are never negative. On the other hand, if an absolute value expression is stated to be less than a negative number, it will never be true, as absolute values cannot be negative. In the following sections, we'll dissect each given inequality to determine whether it holds true for all real numbers, focusing on how these properties come into play.

Detailed Analysis of Each Inequality

Let's delve into each inequality provided, applying our understanding of absolute values to determine which ones have all real numbers as solutions. The goal is to carefully examine how the absolute value expression interacts with the inequality sign and constant terms, paying attention to whether the inequality condition is always met, regardless of the value of x. We will systematically analyze each option, breaking down the steps and logic involved in reaching the conclusion.

A. $-4|6x + 3| < 16$

Our first inequality is $-4|6x + 3| < 16$. To analyze this, we want to isolate the absolute value expression. We can achieve this by dividing both sides of the inequality by -4. It's crucial to remember that when we divide or multiply both sides of an inequality by a negative number, we must reverse the inequality sign. So, dividing by -4 gives us:

$|6x + 3| > -4$

Now, we have the absolute value expression $|6x + 3|$ on one side and -4 on the other. Recall that the absolute value of any expression is always non-negative (greater than or equal to zero). Therefore, $|6x + 3|$ will always be greater than or equal to 0. Since 0 is greater than -4, the inequality $|6x + 3| > -4$ holds true for all real numbers x. No matter what value we substitute for x, the absolute value will never be negative, and thus, it will always be greater than -4.

B. $8|6x + 3| + 13 > 5$

Next, we consider the inequality $8|6x + 3| + 13 > 5$. Again, the strategy is to isolate the absolute value expression. First, we subtract 13 from both sides of the inequality:

$8|6x + 3| > 5 - 13$

$8|6x + 3| > -8$

Now, we divide both sides by 8:

$|6x + 3| > -1$

Similar to the previous case, we have an absolute value expression $|6x + 3|$ being compared to a negative number, -1. Since the absolute value is always non-negative, it will always be greater than -1. Hence, this inequality also holds true for all real numbers x. Regardless of the value of x, $|6x + 3|$ will never be negative, and therefore will always be greater than -1.

C. $|4x - 9| extgreater{} -12$

Moving on to the third inequality, we have $|4x - 9| extgreater{} -12$. This inequality presents a straightforward comparison between an absolute value expression and a negative number. As we've emphasized, the absolute value of any expression is always non-negative. This means $|4x - 9|$ will always be greater than or equal to 0. Since 0 is clearly greater than -12, the inequality $|4x - 9| extgreater{} -12$ will always be true, regardless of the value of x. Therefore, this inequality holds for all real numbers.

D. $-2|x + 1| - 11 extgreater{} 5$

Lastly, let's analyze the inequality $-2|x + 1| - 11 extgreater{} 5$. We start by isolating the absolute value expression. Add 11 to both sides:

$-2|x + 1| extgreater{} 5 + 11$

$-2|x + 1| extgreater{} 16$

Now, divide both sides by -2. Remember to reverse the inequality sign when dividing by a negative number:

$|x + 1| extless{} -8$

Here, we have the absolute value expression $|x + 1|$ being compared to a negative number, -8. However, in this case, the inequality states that the absolute value must be less than -8. This is a contradiction. The absolute value of any expression is always non-negative, meaning it can never be less than a negative number. Therefore, this inequality has no solution, and it certainly does not hold true for all real numbers.

Conclusion: Identifying Inequalities with All Real Solutions

In summary, we've meticulously analyzed each of the given inequalities, leveraging the fundamental property that absolute values are always non-negative. This understanding is crucial in determining whether an absolute value inequality holds true for all real numbers.

We identified that inequalities A, B, and C have all real numbers as their solutions. These inequalities all shared a common characteristic: after isolating the absolute value expression, they showed the absolute value being greater than a negative number. This condition is always satisfied because an absolute value will never be negative.

On the other hand, inequality D presented a contrasting scenario. After isolating the absolute value, it showed the absolute value being less than a negative number, which is impossible. This resulted in inequality D having no solution.

Therefore, the absolute value inequalities that have all real numbers as their solution are:

• $-4|6x + 3| < 16$
• $8|6x + 3| + 13 > 5$
• $|4x - 9| extgreater{} -12$

This analysis highlights the importance of understanding the core properties of absolute values when solving inequalities. By recognizing that absolute values are always non-negative, we can efficiently determine whether an inequality is always true, never true, or true for a specific range of values.

Key Takeaways for Absolute Value Inequalities

To solidify your understanding, let's recap the key principles and strategies we've discussed for solving absolute value inequalities, especially those that result in all real number solutions. These principles will serve as a valuable guide when tackling similar problems in the future.

1. Isolate the Absolute Value Expression: The first and most crucial step is to isolate the absolute value expression on one side of the inequality. This often involves performing algebraic operations such as adding, subtracting, multiplying, or dividing (remembering to reverse the inequality sign when multiplying or dividing by a negative number).
2. Understand the Non-Negativity of Absolute Values: The cornerstone of solving absolute value inequalities lies in the fact that the absolute value of any expression is always non-negative (greater than or equal to zero). This property dictates how absolute values interact with inequalities.
3. Compare to Negative Numbers: Pay close attention when the absolute value expression is compared to a negative number. If the absolute value is stated to be greater than a negative number, the inequality will always hold true for all real numbers, as an absolute value can never be negative. Conversely, if the absolute value is stated to be less than a negative number, the inequality will never hold true and has no solution.
4. Consider Positive Numbers and Zero: When the absolute value expression is compared to a positive number or zero, the inequality might have a specific range of solutions. In these cases, you'll typically need to set up two separate inequalities (one for the positive case and one for the negative case) and solve them individually.
5. Visualize on a Number Line: Sometimes, it can be helpful to visualize the solution on a number line. This is particularly useful when dealing with inequalities that have a specific range of solutions. The number line can provide a visual representation of the values of x that satisfy the inequality.

By mastering these key takeaways, you'll be well-equipped to confidently solve a wide range of absolute value inequalities, including those that have all real numbers as their solutions.

Understanding absolute value inequalities is a vital skill in mathematics, especially when dealing with concepts like distance and error bounds. The core principle to remember is that absolute values are always non-negative, and this property dictates how inequalities involving absolute values behave. By carefully analyzing the structure of the inequality and isolating the absolute value expression, you can efficiently determine whether the solution encompasses all real numbers, no solution, or a specific range of values. Practice is key to mastering these concepts, so continue working through different examples and applying these strategies. With consistent effort, you'll become proficient in solving absolute value inequalities and confidently tackle more advanced mathematical problems.