Solving Q(w) < 0: Inequality Solutions In Interval Notation

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Hey guys! Today, we're diving into solving inequalities, specifically when we're given an inequality in the form of q(w) < 0. This means we're looking for all the values of 'w' that make the expression q(w) less than zero. Basically, where q(w) is negative. To nail this, we'll go through a step-by-step process, ensuring we not only find the solutions but also represent them correctly using interval notation. Interval notation is a neat way to write down sets of numbers, using parentheses and brackets to show whether the endpoints are included or excluded.

Understanding Inequalities

First, let's break down what an inequality is. Unlike equations that have definite solutions (like x = 5), inequalities give us a range of solutions. For example, x > 3 means 'x' can be any number greater than 3. Inequalities can be less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). When we're solving q(w) < 0, we're finding all the 'w' values that make the expression q(w) negative.

Key Concepts to Keep in Mind:

  • Critical Points: These are the values of 'w' where q(w) = 0 or where q(w) is undefined. Critical points are super important because they divide the number line into intervals where q(w) is either positive or negative.
  • Test Intervals: Once we find the critical points, we'll test values within each interval to see if q(w) < 0 in that interval. If it is, then that interval is part of our solution.
  • Interval Notation: We use parentheses (( and )) to show that an endpoint is not included in the solution and brackets ([ and ]) to show that an endpoint is included. For example, (2, 5) means all numbers between 2 and 5, but not including 2 and 5. On the other hand, [2, 5] means all numbers between 2 and 5, including 2 and 5.

Step-by-Step Solution

Alright, let's get into the nitty-gritty of solving q(w) < 0. I'll guide you through each step.

Step 1: Find the Critical Points

To find the critical points, we need to figure out where q(w) = 0 or where q(w) is undefined. This usually involves solving an equation or identifying points where the function has discontinuities (like division by zero). Let's assume, for example, that q(w) = (w - 2)(w + 3). To find where q(w) = 0, we set (w - 2)(w + 3) = 0. This gives us two critical points: w = 2 and w = -3.

Step 2: Create Test Intervals

Now that we have our critical points, we'll use them to divide the number line into intervals. In our example, the critical points w = -3 and w = 2 split the number line into three intervals:

  1. (-∞, -3)
  2. (-3, 2)
  3. (2, ∞)

Step 3: Test Each Interval

Next, we need to pick a test value within each interval and plug it into q(w) to see if the result is negative. Remember, we're looking for where q(w) < 0.

  1. Interval (-∞, -3): Let's pick w = -4. Then, q(-4) = (-4 - 2)(-4 + 3) = (-6)(-1) = 6. Since 6 is not less than 0, this interval is not part of our solution.
  2. Interval (-3, 2): Let's pick w = 0. Then, q(0) = (0 - 2)(0 + 3) = (-2)(3) = -6. Since -6 is less than 0, this interval is part of our solution.
  3. Interval (2, ∞): Let's pick w = 3. Then, q(3) = (3 - 2)(3 + 3) = (1)(6) = 6. Since 6 is not less than 0, this interval is not part of our solution.

Step 4: Write the Solution in Interval Notation

Based on our test intervals, the only interval where q(w) < 0 is (-3, 2). Since the original inequality is strictly less than (not less than or equal to), we don't include the endpoints. Therefore, our solution in interval notation is (-3, 2).

More Examples and Scenarios

Let's tackle a few more examples to solidify our understanding. These will cover different types of functions and inequalities.

Example 1: Rational Inequality

Suppose we have q(w) = (w + 1) / (w - 4) < 0. This is a rational inequality, so we need to be careful about values of 'w' that make the denominator zero.

Step 1: Find Critical Points

  • w + 1 = 0 => w = -1
  • w - 4 = 0 => w = 4 (This is also a point where q(w) is undefined)

Step 2: Create Test Intervals

  1. (-∞, -1)
  2. (-1, 4)
  3. (4, ∞)

Step 3: Test Each Interval

  1. Interval (-∞, -1): Let's pick w = -2. Then, q(-2) = (-2 + 1) / (-2 - 4) = (-1) / (-6) = 1/6. Since 1/6 is not less than 0, this interval is not part of our solution.
  2. Interval (-1, 4): Let's pick w = 0. Then, q(0) = (0 + 1) / (0 - 4) = (1) / (-4) = -1/4. Since -1/4 is less than 0, this interval is part of our solution.
  3. Interval (4, ∞): Let's pick w = 5. Then, q(5) = (5 + 1) / (5 - 4) = (6) / (1) = 6. Since 6 is not less than 0, this interval is not part of our solution.

Step 4: Write the Solution in Interval Notation

The only interval where q(w) < 0 is (-1, 4). Note that we exclude w = 4 because it makes the denominator zero and the function undefined. Thus, our solution is (-1, 4).

Example 2: Quadratic Inequality with ≤

Let's say we have q(w) = w^2 - 5w + 6 ≤ 0. Notice the 'less than or equal to' sign.

Step 1: Find Critical Points

First, factor the quadratic: w^2 - 5w + 6 = (w - 2)(w - 3). Setting this equal to zero gives us w = 2 and w = 3.

Step 2: Create Test Intervals

  1. (-∞, 2)
  2. (2, 3)
  3. (3, ∞)

Step 3: Test Each Interval

  1. Interval (-∞, 2): Let's pick w = 0. Then, q(0) = (0 - 2)(0 - 3) = (-2)(-3) = 6. Since 6 is not less than or equal to 0, this interval is not part of our solution.
  2. Interval (2, 3): Let's pick w = 2.5. Then, q(2.5) = (2.5 - 2)(2.5 - 3) = (0.5)(-0.5) = -0.25. Since -0.25 is less than or equal to 0, this interval is part of our solution.
  3. Interval (3, ∞): Let's pick w = 4. Then, q(4) = (4 - 2)(4 - 3) = (2)(1) = 2. Since 2 is not less than or equal to 0, this interval is not part of our solution.

Step 4: Write the Solution in Interval Notation

Since the inequality is q(w) ≤ 0, we include the endpoints where q(w) = 0. Thus, our solution is [2, 3].

Common Mistakes to Avoid

  • Forgetting to Consider Undefined Points: When dealing with rational inequalities, always remember to check for values that make the denominator zero. These points must be excluded from your solution.
  • Incorrectly Including/Excluding Endpoints: Pay close attention to whether the inequality is strict (<, >) or includes equality (≤, ≥). Strict inequalities use parentheses, while those including equality use brackets.
  • Not Testing Intervals: It's tempting to assume that the solution is simply between the critical points, but you must test each interval to confirm whether it satisfies the inequality.
  • Algebra Errors: Be careful with your algebra! A simple mistake can throw off your entire solution. Double-check your work, especially when factoring or simplifying expressions.

Tips for Success

  • Practice, Practice, Practice: The more you practice, the better you'll become at solving inequalities. Work through a variety of examples with different types of functions.
  • Draw a Number Line: Visualizing the number line with critical points and test intervals can help you keep track of your work and avoid mistakes.
  • Check Your Answer: After finding your solution, pick a value within your interval and plug it back into the original inequality to make sure it holds true.
  • Understand the Concepts: Don't just memorize the steps. Make sure you understand why each step is necessary and how it contributes to finding the solution.

By following these steps and avoiding common mistakes, you'll be well on your way to mastering the art of solving inequalities and expressing your solutions in interval notation. Keep practicing, and you'll become a pro in no time! You got this, guys!