Solving Quadratic Inequalities: Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of quadratic inequalities. Specifically, we're going to tackle the inequality $n^2 - 15n + 50 \geq 0$. Don't worry if this sounds intimidating at first; we'll break it down into manageable steps, making sure you grasp not only the solution but also how to visualize it graphically. This process is crucial because solving quadratic inequalities is a fundamental skill in algebra and has applications in various fields, from physics and engineering to economics and computer science. By understanding this, you're building a strong foundation for more complex mathematical concepts.

First things first, what exactly is a quadratic inequality? Well, it's an inequality that involves a quadratic expression, which is an expression of the form $ax^2 + bx + c$, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. When we talk about inequalities, we're looking for the range of values that satisfy the condition, such as greater than, less than, greater than or equal to, or less than or equal to. In our case, we're dealing with "greater than or equal to". This means we're looking for all the values of 'n' that make the expression $n^2 - 15n + 50$ either positive or equal to zero. The solution will give us the interval or intervals on the number line where this condition holds true. The process involves factoring the quadratic expression, finding the critical points or zeros, and then testing intervals to determine where the inequality is satisfied. It’s a combination of algebraic manipulation and analytical thinking. Now let’s move on to the actual solution of the inequality. We'll start by factoring the quadratic expression.

Step 1: Factoring the Quadratic Expression

Alright, guys, let's start with the first step: factoring the quadratic expression $n^2 - 15n + 50$. Factoring is a super important skill in algebra; it's like the key to unlocking the solutions to many problems. In this case, we need to find two numbers that multiply to 50 and add up to -15. Think about it for a bit. What two numbers fit that description? The numbers we're looking for are -5 and -10. Therefore, the factored form of the expression is $(n - 5)(n - 10)$.

So, our inequality $n^2 - 15n + 50 \geq 0$ becomes $(n - 5)(n - 10) \geq 0$. Now, what does this factored form tell us? It tells us the points where the quadratic expression equals zero are $n = 5$ and $n = 10$. These are the critical points, which are also known as the roots or zeros of the quadratic equation. They are the values of 'n' that make the expression equal to zero. These critical points divide the number line into intervals, and the sign of the expression $(n - 5)(n - 10)$ will either be positive or negative within these intervals. Our next step is to test these intervals to see where our inequality holds true, which means where the expression is greater than or equal to zero. This will allow us to define the solution set for the inequality, which includes all values of 'n' that satisfy the original condition.

Step 2: Finding the Critical Points

Now that we've factored our expression, let's find the critical points. Critical points, as we mentioned earlier, are the values of 'n' that make the expression equal to zero. To find these, we set each factor equal to zero and solve for 'n'.

For the factor $(n - 5)$, we set $n - 5 = 0$. Solving for 'n', we get $n = 5$.

For the factor $(n - 10)$, we set $n - 10 = 0$. Solving for 'n', we get $n = 10$.

So, our critical points are $n = 5$ and $n = 10$. These are the values where the parabola (the graph of our quadratic expression) intersects the x-axis. These points divide the number line into three intervals: $(-\infty, 5)$, $(5, 10)$, and $(10, \infty)$. We'll now test each interval to determine whether the inequality $(n - 5)(n - 10) \geq 0$ is satisfied in that interval. This is an important step to understand because it allows us to identify the regions on the number line where the function is positive or equal to zero. By understanding this, we can accurately determine the solution to the inequality, which represents the set of all 'n' values that fulfill the given condition. We will use these critical points to create a number line and test the inequality within each interval.

Step 3: Testing the Intervals

Time to test those intervals, guys! This is where we figure out which parts of the number line satisfy our inequality. We'll pick a test value from each of the three intervals created by our critical points ($-\infty, 5$), $(5, 10)$, and $(10, \infty$).

Interval 1: $(-\infty, 5)$

Let's choose a test value, say $n = 0$. Plug this into our factored expression: $(0 - 5)(0 - 10) = (-5)(-10) = 50$. Since 50 is greater than 0, the inequality holds true in this interval.

Interval 2: $(5, 10)$

Let's choose a test value, say $n = 7$. Plug this into our factored expression: $(7 - 5)(7 - 10) = (2)(-3) = -6$. Since -6 is not greater than or equal to 0, the inequality does not hold true in this interval.

Interval 3: $(10, \infty)$

Let's choose a test value, say $n = 12$. Plug this into our factored expression: $(12 - 5)(12 - 10) = (7)(2) = 14$. Since 14 is greater than 0, the inequality holds true in this interval.

This testing process is crucial because it helps us to identify the correct solution set for the inequality. We're essentially mapping out where the quadratic expression is positive or equal to zero, which is what our inequality is asking us to find. This method not only helps us solve this specific problem but also provides a general framework for approaching and solving other quadratic inequalities. By understanding this, you can effectively determine which intervals of the number line satisfy the inequality and then accurately represent your solution graphically.

Step 4: Writing the Solution Set

Based on our interval tests, the inequality $(n - 5)(n - 10) \geq 0$ is true for the intervals $(-\infty, 5)$ and $(10, \infty)$. Also, remember that our original inequality includes "equal to," so we must include the critical points where the expression equals zero. Therefore, we include 5 and 10 in our solution set.

So, the solution set is $n \leq 5$ or $n \geq 10$. We can also express this in interval notation as $(-\infty, 5] \cup [10, \infty)$. The square brackets [ ] indicate that the endpoints 5 and 10 are included in the solution because the inequality includes the “equal to” part. Understanding how to express the solution in different formats is essential, as this flexibility allows you to communicate the solution clearly in various contexts. In interval notation, we use parentheses () to indicate that the endpoint is not included, and square brackets [] to indicate that it is included. Combining these intervals correctly accurately describes all the values of 'n' that satisfy the inequality.

Step 5: Graphing the Solution Set

Alright, let’s visualize the solution. Graphing the solution set is a great way to understand what we've found and it's super helpful. To graph it, draw a number line. Mark the critical points, 5 and 10, on the number line. Since our inequality includes "equal to," we'll use closed circles (filled dots) at 5 and 10 to indicate that these points are included in the solution.

Now, shade the regions of the number line that represent our solution set. We know that $n \leq 5$, so we shade the region to the left of 5, going towards negative infinity. Also, we know that $n \geq 10$, so we shade the region to the right of 10, going towards positive infinity. The graph will show two shaded regions, one to the left of 5 and the other to the right of 10, with closed circles at 5 and 10.

This graphical representation makes it easier to understand the solution at a glance. It visually confirms that the inequality holds true for all numbers less than or equal to 5 and all numbers greater than or equal to 10. The graph provides a clear, intuitive understanding of the solution, showing the continuous ranges where the inequality is satisfied. Graphing the solution set solidifies your understanding and offers a visual confirmation of the analytical work you’ve done. It reinforces the concept and provides a clear picture of the possible values of 'n' that satisfy the original inequality. Understanding how to graph inequalities is a fundamental skill in algebra and is crucial for visualizing solutions in other mathematical and scientific contexts.

Conclusion: You've Got This!

And that's a wrap, guys! We've successfully solved the inequality $n^2 - 15n + 50 \geq 0$ and graphed the solution set. We started by factoring the quadratic expression, finding the critical points, testing intervals, and finally, writing and graphing the solution. Remember, practice makes perfect! The more you practice these steps, the easier they will become. Don’t hesitate to try more examples and consult additional resources if you need more practice. Quadratic inequalities are a fundamental part of algebra, and understanding them opens doors to more advanced math concepts. Keep up the great work, and happy solving!