Solving Quadratic Inequality X² - 7x + 6 ≤ 0 A Step By Step Guide

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Hey guys! Today, we're diving deep into the world of quadratic inequalities, specifically focusing on how to solve the inequality x² - 7x + 6 ≤ 0. This type of problem is a classic in algebra, and mastering it will definitely boost your math skills. So, let's break it down step by step, making sure everyone understands the process.

Understanding Quadratic Inequalities

Before we jump into solving, let's quickly recap what quadratic inequalities are all about. Quadratic inequalities are inequalities that involve a quadratic expression, which is a polynomial expression of degree two. They typically take the form of ax² + bx + c > 0, ax² + bx + c < 0, ax² + bx + c ≥ 0, or ax² + bx + c ≤ 0, where a, b, and c are constants, and 'x' is the variable. Solving these inequalities means finding the range (or ranges) of 'x' values that make the inequality true. This is super useful in many real-world applications, from physics to economics, where we often deal with situations involving ranges and constraints.

Key Concepts to Remember

  1. Factoring: Factoring the quadratic expression is often the first step in solving the inequality. It helps us find the critical points.
  2. Critical Points: These are the values of 'x' that make the quadratic expression equal to zero. They are essentially the roots of the quadratic equation.
  3. Test Intervals: After finding the critical points, we divide the number line into intervals. We then test a value from each interval in the original inequality to determine whether the inequality holds true for that interval.
  4. Solution Set: The solution set consists of the intervals where the inequality is true. We express this set using interval notation or inequality notation.

Step-by-Step Solution for x² - 7x + 6 ≤ 0

Okay, let's get our hands dirty and solve the inequality x² - 7x + 6 ≤ 0. We'll go through each step meticulously so you can follow along easily.

Step 1: Factor the Quadratic Expression

The first thing we need to do is factor the quadratic expression x² - 7x + 6. We're looking for two numbers that multiply to 6 and add up to -7. Those numbers are -1 and -6. So, we can factor the expression as follows:

x² - 7x + 6 = (x - 1)(x - 6)

Now our inequality looks like this:

(x - 1)(x - 6) ≤ 0

Step 2: Find the Critical Points

Next, we need to find the critical points. These are the values of 'x' that make the expression (x - 1)(x - 6) equal to zero. We set each factor equal to zero and solve for 'x':

  • x - 1 = 0 => x = 1
  • x - 6 = 0 => x = 6

So, our critical points are x = 1 and x = 6. These points are crucial because they divide the number line into intervals where the expression (x - 1)(x - 6) can change its sign (from positive to negative or vice versa).

Step 3: Create Test Intervals

With the critical points, we now have three intervals to consider:

  1. x < 1
  2. 1 < x < 6
  3. x > 6

We'll pick a test value from each interval and plug it into the inequality (x - 1)(x - 6) ≤ 0 to see if it holds true.

Step 4: Test Each Interval

Let's test each interval:

  1. Interval x < 1:

    • Pick a test value, say x = 0.
    • Plug it into the inequality: (0 - 1)(0 - 6) = (-1)(-6) = 6
    • Since 6 is not less than or equal to 0, this interval is not part of the solution.

  2. Interval 1 < x < 6:

    • Pick a test value, say x = 3.
    • Plug it into the inequality: (3 - 1)(3 - 6) = (2)(-3) = -6
    • Since -6 is less than or equal to 0, this interval is part of the solution.

  3. Interval x > 6:

    • Pick a test value, say x = 7.
    • Plug it into the inequality: (7 - 1)(7 - 6) = (6)(1) = 6
    • Since 6 is not less than or equal to 0, this interval is not part of the solution.

Step 5: Determine the Solution Set

From our testing, we found that the interval 1 < x < 6 makes the inequality true. But we also need to consider the critical points themselves since the inequality is ≤ 0 (less than or equal to zero).

  • At x = 1, (1 - 1)(1 - 6) = 0, which satisfies the inequality.
  • At x = 6, (6 - 1)(6 - 6) = 0, which also satisfies the inequality.

Therefore, the solution set includes the interval between 1 and 6, as well as the points 1 and 6 themselves. We can write this in interval notation as [1, 6].

Conclusion

So, the solution set for the quadratic inequality x² - 7x + 6 ≤ 0 is 1 ≤ x ≤ 6. This corresponds to option C in your list. You see, guys, by breaking down the problem into manageable steps, we can easily solve even complex inequalities!

Common Mistakes to Avoid

Before we wrap up, let's talk about some common mistakes people make when solving quadratic inequalities. Avoiding these will save you a lot of headaches!

  1. Forgetting to Consider Critical Points:

    • It's easy to get caught up in testing intervals and forget to include the critical points themselves in the solution, especially when the inequality is ≤ or ≥. Always check if the critical points satisfy the inequality.

  2. Incorrectly Factoring the Quadratic Expression:

    • Factoring is a crucial step, and a mistake here can throw off the entire solution. Double-check your factoring to make sure it's correct.

  3. Mixing Up the Signs:

    • When testing intervals, pay close attention to the signs. A simple sign error can lead to the wrong conclusion about whether an interval is part of the solution.

  4. Not Understanding the Inequality Symbols:

    • Make sure you understand what the inequality symbols (<, >, ≤, ≥) mean. For example, ≤ means “less than or equal to,” so you need to include the points where the expression equals zero.

Practice Problems

To really nail this down, practice is key! Here are a few more quadratic inequalities you can try solving:

  1. x² - 5x + 4 > 0
  2. 2x² + 3x - 2 ≤ 0
  3. x² - 9 ≥ 0

Work through these problems using the steps we discussed, and you'll become a pro at solving quadratic inequalities in no time!

Real-World Applications

Now, you might be wondering, “Where do we actually use this stuff?” Well, quadratic inequalities pop up in various real-world scenarios.

  1. Physics: In physics, you might use quadratic inequalities to determine the range of initial velocities required for a projectile to reach a certain height.
  2. Engineering: Engineers use them in structural design to ensure that stress and strain levels stay within safe limits.
  3. Economics: In economics, quadratic inequalities can help model profit margins and cost-benefit analyses.
  4. Optimization Problems: Many optimization problems, where you're trying to find the maximum or minimum value of something, involve quadratic inequalities.

For instance, imagine a company wants to maximize its profit. The profit function might be a quadratic equation, and the company needs to find the range of production levels that yield a profit above a certain threshold. This is where solving quadratic inequalities comes in handy.

Advanced Tips and Tricks

For those of you who want to take your skills to the next level, here are a few advanced tips and tricks:

  1. Using the Quadratic Formula: If the quadratic expression is difficult to factor, you can always use the quadratic formula to find the roots (critical points). The quadratic formula is given by:

    • x = (-b ± √(b² - 4ac)) / (2a)

  2. Graphing the Quadratic Function: Graphing the quadratic function can provide a visual representation of the solution. The intervals where the graph is above or below the x-axis correspond to the solution of the inequality.

  3. Completing the Square: Completing the square can be another useful technique for solving quadratic inequalities, especially when the expression is not easily factored.

By mastering these techniques, you'll be well-equipped to tackle even the most challenging quadratic inequality problems. Keep practicing, and don't be afraid to explore different approaches. Math is all about problem-solving, and with the right tools and mindset, you can conquer any challenge!

So, there you have it, guys! A comprehensive guide to solving the quadratic inequality x² - 7x + 6 ≤ 0. Remember, the key is to break it down, understand each step, and practice, practice, practice. Keep up the great work, and I'll see you in the next math adventure!