Solving (x+5)(x+3)(x-7) ≤ 0: Inequality Solution

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Alright, let's dive into solving this inequality: (x+5)(x+3)(x-7) ≤ 0. Inequalities like these pop up all the time in mathematics, and mastering them is super useful. We're going to break it down step-by-step so you can tackle similar problems with confidence. Our goal is to find all the values of x that make this statement true. We will express our answer in interval notation. So, grab your pencils, and let’s get started!

1. Find the Critical Points

The first thing we need to do is find the critical points. These are the values of x that make the expression equal to zero. To find them, we set each factor equal to zero and solve for x:

  • x + 5 = 0 => x = -5
  • x + 3 = 0 => x = -3
  • x - 7 = 0 => x = 7

So, our critical points are x = -5, -3, and 7. These points are crucial because they divide the number line into intervals where the expression (x+5)(x+3)(x-7) will either be positive or negative. Basically, these are the turning points where the expression can change signs.

2. Create a Sign Chart

Now that we have our critical points, we'll create a sign chart. This chart helps us determine the sign of the expression in each interval. We'll place our critical points on the number line and test a value from each interval in the expression (x+5)(x+3)(x-7).

Here’s how we set up the sign chart:

  • Draw a number line and mark the critical points: -5, -3, and 7.
  • These points divide the number line into four intervals: (-∞, -5), (-5, -3), (-3, 7), and (7, ∞).

Now, we pick a test value from each interval and plug it into our expression:

  • Interval (-∞, -5): Let's pick x = -6
    • (-6 + 5)(-6 + 3)(-6 - 7) = (-1)(-3)(-13) = -39 (Negative)
  • Interval (-5, -3): Let's pick x = -4
    • (-4 + 5)(-4 + 3)(-4 - 7) = (1)(-1)(-11) = 11 (Positive)
  • Interval (-3, 7): Let's pick x = 0
    • (0 + 5)(0 + 3)(0 - 7) = (5)(3)(-7) = -105 (Negative)
  • Interval (7, ∞): Let's pick x = 8
    • (8 + 5)(8 + 3)(8 - 7) = (13)(11)(1) = 143 (Positive)

So, the sign chart looks like this:

  • (-∞, -5): Negative
  • (-5, -3): Positive
  • (-3, 7): Negative
  • (7, ∞): Positive

3. Determine the Solution

We want to find where (x+5)(x+3)(x-7) ≤ 0. This means we are looking for the intervals where the expression is negative or equal to zero. From our sign chart, we can see that the expression is negative in the intervals (-∞, -5) and (-3, 7).

Since we also want to include where the expression is equal to zero, we include the critical points in our solution. Therefore, the solution is:

(-∞, -5] ∪ [-3, 7]

This is the union of two intervals: from negative infinity to -5 (inclusive) and from -3 to 7 (inclusive). This means that any value of x in these intervals will satisfy the original inequality. This comprehensive approach ensures no detail is overlooked.

4. Expressing the Solution Set

In summary, the solution to the inequality (x+5)(x+3)(x-7) ≤ 0 is expressed as the union of two intervals: (-∞, -5] and [-3, 7]. Graphically, this represents all real numbers less than or equal to -5, as well as all real numbers between -3 and 7, including -3 and 7 themselves. Understanding the solution set helps in visualizing and interpreting the range of values that satisfy the given condition. This approach provides clarity and facilitates practical applications of the solution in various contexts.

Visual Representation

A visual representation of the number line can further enhance understanding. Imagine a number line stretching from negative infinity to positive infinity. The critical points -5, -3, and 7 are marked on this line. The intervals (-∞, -5] and [-3, 7] are shaded, indicating that these are the regions where the inequality holds true. This visual aid offers a quick and intuitive way to grasp the solution set and its implications.

Practical Implications

The solution to this inequality has practical implications in various fields, such as engineering, physics, and economics. For example, in engineering, understanding the range of values that satisfy certain conditions can help in designing stable and efficient systems. In economics, it can be used to model scenarios where profitability is dependent on certain parameters falling within specific ranges. The ability to solve inequalities like this one is a valuable skill in quantitative analysis and decision-making.

Common Mistakes to Avoid

When solving inequalities, it's important to avoid common mistakes that can lead to incorrect solutions. One common mistake is forgetting to include the critical points in the solution set when the inequality includes an "equal to" condition (≤ or ≥). Another mistake is incorrectly determining the sign of the expression in each interval. Always double-check your calculations and ensure that you are using the correct test values. Additionally, be careful when dealing with inequalities involving rational expressions or absolute values, as these require special considerations.

Tips for Success

To improve your ability to solve inequalities, practice is key. Work through a variety of examples, starting with simpler inequalities and gradually progressing to more complex ones. Pay attention to detail and double-check your work. Use sign charts to organize your thoughts and avoid mistakes. Additionally, seek out resources such as textbooks, online tutorials, and practice problems to reinforce your understanding. With consistent effort and attention to detail, you can master the art of solving inequalities and apply this skill to various mathematical and real-world problems.

Conclusion

So, there you have it, guys! The solution to the inequality (x+5)(x+3)(x-7) ≤ 0 is (-∞, -5] ∪ [-3, 7]. Breaking down the problem into steps—finding critical points, creating a sign chart, and determining the solution—makes it much easier to handle. Keep practicing, and you'll become a pro at solving these types of inequalities. Happy problem-solving!