Solving Absolute Value Inequalities Algebraically

Let's dive into the world of absolute value inequalities! If you're scratching your head trying to figure out how to solve them, you're in the right place. This article will guide you through the steps to algebraically solve an inequality like $-2|9-3x|-5 \leq -17$. We'll break it down piece by piece, so you'll not only understand the process but also feel confident tackling similar problems. So, grab your pen and paper, and let’s get started!

Understanding Absolute Value

Before we jump into solving the inequality, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. This means it's always non-negative. For example, $|3| = 3$ and $|-3| = 3$. Understanding this concept is crucial because absolute value expressions often lead to two different cases when solving inequalities.

When dealing with absolute value inequalities, it's essential to remember that the expression inside the absolute value can be either positive or negative, but its distance from zero remains the same. This dual nature is what leads to the need for considering two separate cases in the solution process. Grasping this fundamental idea will make solving these types of inequalities much more straightforward and less intimidating. We’ll see how this plays out as we walk through the steps of solving our specific inequality.

The Initial Inequality: -2|9-3x|-5 old{\leq} -17

Our mission is to solve the inequality $-2|9-3x|-5 \leq -17$. The key here is to isolate the absolute value expression first. Think of it like solving a regular algebraic equation where you need to get the variable by itself. We'll apply similar principles here, but with the added consideration of the absolute value.

The initial step in solving this absolute value inequality is crucial, as it sets the stage for the subsequent steps. Isolating the absolute value term is akin to preparing the equation for further analysis. This process involves performing algebraic manipulations to ensure that the absolute value expression stands alone on one side of the inequality. This isolation allows us to then address the two possible scenarios that arise due to the nature of absolute values—the positive and negative cases. By methodically isolating the absolute value, we streamline the solving process and reduce the likelihood of errors, paving the way for an accurate solution.

Step 1: Isolate the Absolute Value

To isolate the absolute value, we need to get the term $|9-3x|$ by itself. First, let's add 5 to both sides of the inequality:

$-2|9-3x|-5 + 5 \leq -17 + 5$

This simplifies to:

$-2|9-3x| \leq -12$

Next, we'll divide both sides by -2. Remember, when you divide or multiply an inequality by a negative number, you must flip the inequality sign:

$\frac{-2|9-3x|}{-2} \geq \frac{-12}{-2}$

This gives us:

$|9-3x| \geq 6$

Isolating the absolute value is a critical step because it allows us to clearly see the condition we need to address. By getting $|9-3x|$ alone on one side, we can now focus on the two possibilities: either the expression inside the absolute value, $9-3x$, is greater than or equal to 6, or it is less than or equal to -6. This branching into two cases is a direct consequence of the absolute value, which disregards the sign of the expression inside it. With the absolute value isolated, we can systematically explore these two scenarios, ensuring that we capture all possible solutions to the inequality.

Step 2: Split into Two Cases

Now that we have the absolute value isolated, $|9-3x| \geq 6$, we need to consider two cases:

Case 1: The expression inside the absolute value is greater than or equal to 6:

$9-3x \geq 6$

Case 2: The expression inside the absolute value is less than or equal to -6:

$9-3x \leq -6$

Splitting the absolute value inequality into two distinct cases is where the unique nature of absolute value truly comes into play. This step acknowledges that the absolute value of an expression being greater than or equal to a certain number implies two possibilities: either the expression itself is greater than or equal to that number, or the expression is less than or equal to the negative of that number. Failing to consider both cases would lead to an incomplete solution set. By carefully bifurcating the problem, we ensure that all possible values of x that satisfy the original inequality are accounted for.

Step 3: Solve Case 1: 9-3x old{\geq} 6

Let's solve the first case, $9-3x \geq 6$. First, subtract 9 from both sides:

$9-3x - 9 \geq 6 - 9$

$-3x \geq -3$

Now, divide both sides by -3. Remember to flip the inequality sign:

$\frac{-3x}{-3} \leq \frac{-3}{-3}$

$x \leq 1$

In solving Case 1, the algebraic manipulations are straightforward but require careful attention to the rules of inequalities. Subtracting 9 from both sides isolates the term with x, and then dividing by -3, we must remember to reverse the inequality sign. This step is crucial because failing to flip the sign would lead to an incorrect solution set for this case. The result, $x \leq 1$, represents one part of the overall solution to the original absolute value inequality. This highlights the importance of meticulous execution in each step to ensure an accurate final solution.

Step 4: Solve Case 2: 9-3x old{\leq} -6

Now let's tackle the second case, $9-3x \leq -6$. Again, subtract 9 from both sides:

$9-3x - 9 \leq -6 - 9$

$-3x \leq -15$

Divide both sides by -3, and remember to flip the inequality sign:

$\frac{-3x}{-3} \geq \frac{-15}{-3}$

$x \geq 5$

Solving Case 2 mirrors the process of Case 1 but yields a different part of the solution set. The steps involve similar algebraic manipulations—subtracting 9 and then dividing by -3, again being mindful to reverse the inequality sign. The result, $x \geq 5$, provides the second piece of the solution to the original absolute value inequality. Together with the solution from Case 1, we are beginning to form a comprehensive understanding of the values of x that satisfy the initial inequality. This systematic approach of solving each case individually ensures that no potential solutions are overlooked.

Step 5: Combine the Solutions

We've found two sets of solutions:

$x \leq 1$ or $x \geq 5$

This means that any value of $x$ that is less than or equal to 1, or greater than or equal to 5, will satisfy the original inequality.

Combining the solutions from both cases is the final step in solving the absolute value inequality. The solution $x \leq 1$ represents all values of x that are less than or equal to 1, while $x \geq 5$ represents all values of x that are greater than or equal to 5. These two sets of values together form the complete solution set. In mathematical terms, we express this combined solution using the “or” conjunction, indicating that x can belong to either of these intervals. This final solution encapsulates all possible values of x that make the original inequality true.

Step 6: Express the Solution in Interval Notation (Optional)

For those who prefer interval notation, we can express the solution as:

$(-\infty, 1] \cup [5, \infty)$

This notation clearly shows the two intervals where the solutions lie.

Expressing the solution in interval notation provides a concise and standardized way to represent the range of values that satisfy the inequality. The notation $(-\infty, 1]$ represents all real numbers less than or equal to 1, while $[5, \infty)$ represents all real numbers greater than or equal to 5. The symbol $\cup$ indicates the union of these two intervals, meaning that the solution set includes all numbers in either interval. This notation is particularly useful in higher-level mathematics and provides a clear visual representation of the solution set on the number line.

Conclusion

Solving absolute value inequalities might seem tricky at first, but by breaking the problem down into steps—isolating the absolute value, splitting into cases, solving each case, and combining the solutions—it becomes manageable. Remember to flip the inequality sign when dividing or multiplying by a negative number, and always consider both positive and negative cases for the expression inside the absolute value. With practice, you'll become a pro at solving these types of inequalities!

For more practice and a deeper understanding of inequalities, you might find valuable resources on websites like Khan Academy. They offer a variety of lessons and practice problems to help you master this topic.