Solving (x-17)(x+17)/x ≤ 0: A Step-by-Step Guide

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Hey guys! Today, we're diving into how to solve the inequality (x-17)(x+17)/x ≤ 0 algebraically. This might look intimidating at first, but trust me, we'll break it down into manageable steps. We'll explore each part of the problem, ensuring you understand not just the how, but also the why behind each step. So, let's get started and make this inequality a piece of cake!

1. Understanding the Inequality

Before we jump into solving, let's make sure we understand what the inequality (x-17)(x+17)/x ≤ 0 is telling us. Essentially, we're looking for all the values of x that make the expression on the left side less than or equal to zero. This means the expression can be negative or zero. The key here is the interplay between the numerator (x-17)(x+17) and the denominator x. Their signs will determine the overall sign of the expression. Remember, a fraction is negative if the numerator and denominator have opposite signs, and it's positive if they have the same sign. We're also looking for where the expression equals zero, which happens when the numerator is zero (but the denominator isn't, because division by zero is a big no-no!). Understanding this foundational concept is crucial because it guides our entire problem-solving approach. We're not just manipulating symbols; we're figuring out where a mathematical expression is behaving in a certain way. This understanding will be our compass as we navigate through the algebraic steps.

2. Finding the Critical Values

The first crucial step in solving this inequality is identifying the critical values. These are the values of x that make either the numerator or the denominator of our expression equal to zero. Why are these values so important? Because they are the points where the expression can change its sign. Think of it like this: the critical values are the potential "switch points" between positive and negative regions on the number line. To find these switch points, we need to set each factor in the numerator and the denominator equal to zero and solve for x. This gives us a clear picture of where the expression might transition from being positive to negative, or vice versa. These critical values are the anchors around which we will build our solution. We are essentially mapping out the landscape of the inequality, and these critical values are the landmarks that define the terrain. By finding them, we're setting the stage for a systematic analysis of the inequality's behavior.

Setting the Numerator to Zero

Let's start by setting the numerator, (x-17)(x+17), equal to zero. This gives us the equation (x-17)(x+17) = 0. To solve this, we use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, either x-17 = 0 or x+17 = 0. Solving these simple equations, we find that x = 17 and x = -17. These are two of our critical values, the points where the expression can potentially equal zero.

Setting the Denominator to Zero

Next, we need to consider the denominator, x. Setting x equal to zero gives us x = 0. This is another critical value, but it's a special one. While the values from the numerator made the entire expression equal to zero, this value makes the expression undefined. Remember, division by zero is not allowed in mathematics! So, x = 0 is a critical value where the function doesn't exist, which is just as important to consider as where it equals zero.

3. Creating a Sign Chart

Now that we've identified our critical values (-17, 0, and 17), the next step is to create a sign chart. A sign chart is a visual tool that helps us determine the sign of our expression, (x-17)(x+17)/x, in different intervals along the number line. Think of it as a map that shows us where the expression is positive, negative, or zero. The critical values act as dividers, splitting the number line into distinct intervals. Within each interval, the sign of the expression will remain constant because there are no other critical values to cause a change. This is the power of the sign chart – it allows us to analyze the expression's behavior in a structured way, interval by interval.

Dividing the Number Line

Our critical values, -17, 0, and 17, divide the number line into four intervals: (-∞, -17), (-17, 0), (0, 17), and (17, ∞). These intervals represent all possible values of x, and within each of these intervals, our expression will have a consistent sign (either positive or negative). The sign chart helps us visualize these intervals and the sign of the expression within each one.

Testing Values in Each Interval

To determine the sign of the expression in each interval, we'll choose a test value within that interval and plug it into our expression, (x-17)(x+17)/x. We don't need to calculate the exact value; we only care about the sign (positive or negative). Let's walk through each interval:

  1. Interval (-∞, -17): Let's choose a test value, say x = -18. Plugging this into our expression gives us ((-18)-17)((-18)+17)/(-18) = (-35)(-1)/(-18), which is negative.
  2. Interval (-17, 0): Let's choose x = -1. Plugging this in gives us ((-1)-17)((-1)+17)/(-1) = (-18)(16)/(-1), which is positive.
  3. Interval (0, 17): Let's choose x = 1. Plugging this in gives us ((1)-17)((1)+17)/(1) = (-16)(18)/(1), which is negative.
  4. Interval (17, ∞): Let's choose x = 18. Plugging this in gives us ((18)-17)((18)+17)/(18) = (1)(35)/(18), which is positive.

Constructing the Sign Chart Table

Now, let's organize our findings into a sign chart table. This table will clearly show the intervals, test values, and the sign of the expression in each interval. It's a powerful visual aid that makes it easy to see the solution.

Interval Test Value (x-17) (x+17) x (x-17)(x+17)/x Sign
(-∞, -17) x = -18 - - - - -
(-17, 0) x = -1 - + - + +
(0, 17) x = 1 - + + - -
(17, ∞) x = 18 + + + + +

4. Determining the Solution

Now for the exciting part – figuring out the solution to our inequality! Remember, we're looking for the intervals where (x-17)(x+17)/x ≤ 0. This means we want the intervals where the expression is either negative or equal to zero. Our sign chart is the key to unlocking this information. It clearly shows us where the expression is negative, and we already know where it equals zero (at the critical values from the numerator). Let's piece together the solution by examining our sign chart and considering the meaning of the inequality symbol.

Identifying Negative Intervals

Looking at our sign chart, we can see that the expression (x-17)(x+17)/x is negative in the intervals (-∞, -17) and (0, 17). These are the intervals where the expression is strictly less than zero.

Including Zero Values

Our inequality also includes "or equal to zero" (≤), so we need to consider the values of x that make the expression equal to zero. These are the critical values that came from the numerator: x = -17 and x = 17. We include these values in our solution by using square brackets in our interval notation.

Excluding Undefined Values

However, we need to be careful about the critical value x = 0. This value makes the denominator zero, and the expression is undefined at this point. So, we cannot include x = 0 in our solution. This is a crucial distinction because including it would be mathematically incorrect. We use parentheses around 0 in our interval notation to indicate that it's not included.

Writing the Final Solution

Putting it all together, our solution includes the intervals where the expression is negative and the points where it equals zero, while excluding the point where it's undefined. In interval notation, the solution to the inequality (x-17)(x+17)/x ≤ 0 is (-∞, -17] ∪ (0, 17]. This means that any value of x within these intervals will satisfy the original inequality. Woo-hoo! We did it!

5. Visualizing the Solution on a Number Line

To really solidify our understanding, let's visualize the solution on a number line. This provides a clear and intuitive representation of the values of x that satisfy our inequality. A number line is a simple but powerful tool for understanding inequalities.

Marking Critical Values

First, we draw a number line and mark our critical values: -17, 0, and 17. These points are the key landmarks that define our solution intervals. They divide the number line into the regions we've already analyzed.

Representing Intervals

Next, we represent the intervals in our solution. For the interval (-∞, -17], we draw a line extending from negative infinity up to -17. Since -17 is included in the solution (because of the "or equal to" part of the inequality), we use a closed circle or a square bracket at -17 to indicate inclusion. For the interval (0, 17], we draw a line extending from 0 to 17. Since 0 is not included (because it makes the expression undefined), we use an open circle or a parenthesis at 0 to indicate exclusion. At 17, we use a closed circle or a square bracket to show that it is included.

Shading the Solution

Finally, we shade the regions of the number line that represent our solution intervals. This visually highlights the values of x that satisfy the inequality. The shaded regions represent all the numbers that, when plugged into the original inequality, will make it true. This visual representation can be incredibly helpful in understanding the scope and nature of the solution.

Conclusion

So, there you have it! We've successfully solved the inequality (x-17)(x+17)/x ≤ 0 algebraically. We started by understanding the inequality, found the critical values, created a sign chart, determined the solution, and even visualized it on a number line. This step-by-step approach can be applied to many other inequalities, making you a true inequality-solving pro! Remember, the key is to break down complex problems into smaller, manageable steps, and always understand the why behind each step. Keep practicing, and you'll master these skills in no time. You got this!