Unveiling Solutions: The Inequality 4b² + 14b + 16 < 10 Explained

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Hey guys! Let's dive into solving the quadratic inequality $4b^2 + 14b + 16 < 10$. This might seem a bit intimidating at first, but trust me, we'll break it down into easy-to-understand steps. Our goal is to figure out the range of values for 'b' that make this inequality true. We'll explore the concepts, ensuring you grasp the process thoroughly. Think of it as a journey where we explore and solve a problem, from the initial setup to the final answer. We'll start with the basics, then gradually move towards the solution. So, grab your pencils and let's get started!

Step-by-Step Guide to Solving the Inequality

Alright, let's roll up our sleeves and solve the inequality $4b^2 + 14b + 16 < 10$. The main goal is to find the values of 'b' that satisfy this condition. Here’s a breakdown of the steps:

  1. Rearrange the Inequality: The first thing we need to do is get everything on one side of the inequality and have a zero on the other side. This helps us work with the quadratic expression more effectively. So, let's subtract 10 from both sides:

    4b2+14b+16−10<10−104b^2 + 14b + 16 - 10 < 10 - 10

    4b2+14b+6<04b^2 + 14b + 6 < 0

  2. Simplify if Possible: Before we move on, let's see if we can simplify the quadratic expression. We can actually divide the entire inequality by 2 to make the numbers smaller and easier to handle:

    (4b2+14b+6)/2<0/2(4b^2 + 14b + 6) / 2 < 0 / 2

    2b2+7b+3<02b^2 + 7b + 3 < 0

  3. Find the Roots: Now, we need to find the roots (or zeros) of the quadratic equation $2b^2 + 7b + 3 = 0$. These roots are the values of 'b' where the quadratic expression equals zero. We can use factoring, the quadratic formula, or completing the square to find these roots. In this case, factoring works nicely. We're looking for two numbers that multiply to give us (2 * 3 = 6) and add up to 7. Those numbers are 6 and 1. So, we can rewrite and factor the equation:

    2b2+6b+b+3=02b^2 + 6b + b + 3 = 0

    2b(b+3)+1(b+3)=02b(b + 3) + 1(b + 3) = 0

    (2b+1)(b+3)=0(2b + 1)(b + 3) = 0

    Therefore, the roots are:

    2b + 1 = 0 => b = -1/2$ and $b + 3 = 0 => b = -3

  4. Plot the Roots on a Number Line: Next, draw a number line and mark the roots we just found (-3 and -1/2). These roots divide the number line into intervals. Since the inequality is strictly less than (<), we'll use open circles at the roots to indicate that these values are not included in the solution.

  5. Test Intervals: We now need to test the intervals created by the roots to determine where the quadratic expression $2b^2 + 7b + 3$ is less than zero. Pick a test value from each interval and plug it into the expression:

    • Interval 1: b < -3: Let's test b = -4.

      2(-4)^2 + 7(-4) + 3 = 32 - 28 + 3 = 7$ (Positive)

    • Interval 2: -3 < b < -1/2: Let's test b = -1.

      2(-1)^2 + 7(-1) + 3 = 2 - 7 + 3 = -2$ (Negative)

    • Interval 3: b > -1/2: Let's test b = 0.

      2(0)^2 + 7(0) + 3 = 3$ (Positive)

  6. Determine the Solution: The quadratic expression is negative (less than zero) only in the interval between -3 and -1/2. Therefore, the solution to the inequality $2b^2 + 7b + 3 < 0$ is $-3 < b < -1/2$. This means that any value of 'b' between -3 and -1/2 will satisfy the original inequality.

So, there you have it, folks! We've successfully solved the inequality. The key is to take it step by step, understand each process, and you'll get the hang of it.

Detailed Explanation: Understanding the Concepts

Alright, let's break down the concepts behind solving the quadratic inequality. Understanding these concepts helps you not just solve this particular problem but also tackle similar problems with confidence. It all starts with the basics of quadratic expressions, roots, and inequalities.

  • Quadratic Expressions: A quadratic expression is an expression of the form $ax^2 + bx + c$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These expressions create parabolas when graphed. The roots (or zeros) of a quadratic expression are the points where the parabola intersects the x-axis (where the value of the expression equals zero).

  • Roots of a Quadratic Equation: The roots are the solutions to the quadratic equation $ax^2 + bx + c = 0$. They can be found using various methods such as factoring, the quadratic formula, or completing the square. The roots are crucial because they divide the number line into intervals where the quadratic expression is either positive or negative.

  • Inequalities: Inequalities compare two values, indicating whether one is less than, greater than, less than or equal to, or greater than or equal to the other. Solving a quadratic inequality involves finding the range of values for the variable that satisfies the inequality. This often involves determining the intervals where the quadratic expression is positive or negative.

Now, let's connect these concepts to our specific problem, $4b^2 + 14b + 16 < 10$. We've seen that the quadratic expression can be rearranged and simplified to $2b^2 + 7b + 3 < 0$. When we find the roots of the corresponding quadratic equation $2b^2 + 7b + 3 = 0$, we are essentially finding the x-intercepts of the parabola represented by the quadratic expression. These x-intercepts are at b = -3 and b = -1/2. The x-intercepts (roots) are used to split the number line into intervals. The sign of the quadratic expression (positive or negative) within each interval is determined by testing a value in that interval. Finally, the solution to the inequality is the interval(s) where the quadratic expression satisfies the inequality condition (in this case, less than zero).

Let’s emphasize again why each step is critical.

  • Rearranging the Inequality: ensures the comparison is made against zero, allowing us to easily determine when the expression is positive or negative.
  • Finding the Roots: These points define the boundaries (endpoints) of the solution intervals.
  • Testing Intervals: This step determines which intervals satisfy the inequality. This is the heart of the process, identifying the regions where the quadratic expression matches the given inequality, making it the most important step for you to understand fully.

The Importance of Understanding the Concepts

Understanding the theory behind solving quadratic inequalities is crucial. It’s not just about memorizing steps, but understanding why you're performing each step. This approach allows you to solve a wider range of problems, even those that look different at first glance.

  • Conceptual Understanding: Knowing why you're doing each step helps you adapt to different problems and identify errors more easily. When you understand the underlying principles, you're better equipped to adjust your strategy as needed.

  • Problem-Solving Skills: You'll build stronger problem-solving skills, applying the same method to various types of inequalities or mathematical problems.

  • Error Detection: You'll be able to spot mistakes more readily because you'll recognize whether an answer makes sense in the context of the problem.

So, whether you're working on a homework assignment, preparing for an exam, or just trying to brush up on your algebra skills, mastering the concepts will serve you well. Take time to revisit each step, and make sure you truly understand why you are doing it, and your confidence will increase, helping you ace problems like these.

Visualizing the Solution: Graphical Representation

Let's get visual! Plotting the quadratic function $y = 2b^2 + 7b + 3$ will give us a clear picture of the solution to the inequality $2b^2 + 7b + 3 < 0$. Visualizing the solution through a graphical representation makes it much easier to understand the solution.

  1. Graphing the Parabola: The quadratic function $y = 2b^2 + 7b + 3$ is a parabola. The roots we found earlier (-3 and -1/2) are the x-intercepts of this parabola. That's where the parabola crosses the x-axis.

  2. Identifying the Solution Region: The inequality $2b^2 + 7b + 3 < 0$ asks us where the function is less than zero. In terms of the graph, this means we want to find the section of the parabola that lies below the x-axis (where y < 0). If you sketch the parabola, the segment below the x-axis lies between the roots -3 and -1/2.

  3. Interpreting the Graph: You can see that when $-3 < b < -1/2$, the parabola is below the x-axis, which confirms our solution. The graph provides a direct visual confirmation of your answer.

This is why graphical representations are so important in math. They show the relationships between mathematical concepts in a clear, easy-to-understand way.

Common Mistakes and How to Avoid Them

Alright, let's talk about some common pitfalls when solving quadratic inequalities like $4b^2 + 14b + 16 < 10$ and how to steer clear of them. Recognizing and avoiding these mistakes will help you solve problems more accurately and efficiently. Don’t worry; we all make mistakes, but learning from them is how we improve.

  1. Forgetting to Rearrange the Inequality: One of the most common errors is not getting the inequality into the standard form. Remember, the inequality must be in the form of $ax^2 + bx + c < 0$, $ax^2 + bx + c > 0$, $ax^2 + bx + c ≤ 0$, or $ax^2 + bx + c ≥ 0$. This is because we need to compare the quadratic expression to zero to determine where it is positive or negative. The solution interval's boundaries are found using the roots of the corresponding quadratic equation. Always bring all terms to one side of the inequality. Subtracting 10 from both sides in our original equation, $4b^2 + 14b + 16 < 10$ , is essential for this. Don't skip this step!

  2. Incorrect Factoring: If you're solving the quadratic equation by factoring, make sure the factors are correct. Double-check that your factored form multiplies back to the original quadratic expression. Using the wrong factors leads to the wrong roots, and then to the wrong solution. Take your time, and if factoring looks tricky, use the quadratic formula.

  3. Misinterpreting the Number Line: When testing the intervals, be careful about the signs. A common mistake is to pick the wrong test values or to misinterpret the intervals. Draw a clear number line and mark the roots accurately. Then, pick a test value from each interval and check its sign. Remember that the sign of the quadratic expression changes at the roots.

  4. Forgetting to Consider the Inequality's Direction: If the original inequality is $<$ or $>$ you should use open circles on the number line to represent the roots because these points are not included in the solution. If the inequality is $\leq$ or $\geq$, then you should use closed circles, because the roots are included in the solution. Make sure you match the inequality sign with the correct type of circle when representing the solution graphically.

  5. Not Checking the Solution: Always, always, always check your solution. Pick a test value in the solution interval (and one outside of it) and plug it back into the original inequality. This helps you confirm that your solution is correct. If the inequality holds true, you're golden! If not, review your steps to find the error.

By keeping these common mistakes in mind, you will prevent them and improve your ability to solve quadratic inequalities.

Conclusion: Putting It All Together

We've covered a lot of ground, guys. We started with the inequality $4b^2 + 14b + 16 < 10$ and went through each step to find the solution. We rearranged, simplified, found the roots, tested intervals, and even visualized the solution with a graph. Remember the fundamental steps to solve such a problem. First, rearrange to form a standard inequality, find the roots, and then test the intervals. Finally, always double-check your answers. Keep practicing, and you'll find that solving quadratic inequalities becomes easier and more intuitive over time. Keep going, and you'll become a pro at these problems in no time! Keep practicing, and you'll become a pro at these problems in no time!