Solving Inequalities: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of inequalities, specifically tackling the problem of how to solve the inequality ${\frac{1}{x+3}<2}$ and express the solution in interval notation. Don't worry, it might seem a bit tricky at first, but we'll break it down step-by-step to make sure everyone understands. Inequalities are super important in math; they pop up in all sorts of real-world scenarios, from figuring out the best deal on something to understanding how things change over time. So, let's get started and learn how to solve them like pros!

Understanding the Basics of Inequalities

Alright, before we jump into solving ${\frac{1}{x+3}<2}$, let's quickly recap what inequalities are all about. Basically, inequalities are mathematical statements that compare two values, showing that one is less than, greater than, less than or equal to, or greater than or equal to another. We use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to represent these relationships. Understanding these symbols is key to correctly interpreting and solving inequalities. For instance, when we see ${x < 5}$, it means any value of ${x}$ that's smaller than 5 is a solution. When we solve an inequality, our goal is to find all the values of the variable that make the statement true. This is often represented using interval notation or graphically on a number line. Keep in mind that solving inequalities often involves similar steps to solving equations, but there's a critical difference when multiplying or dividing by a negative number. When you do that, you have to flip the inequality sign! It's a small detail, but it makes a big difference in getting the right answer. Furthermore, remember that the solution to an inequality is not a single number, but a range of numbers. This range represents all the values that satisfy the given inequality. Mastering these fundamental concepts is super important for not only solving this specific problem, but also for tackling more complex mathematical challenges in the future. So, let's keep these rules in mind as we work through the problem.

Now, let's talk about the specific problem we're solving today: ${\frac{1}{x+3}<2}$. This inequality involves a fraction, which means we'll need to be extra careful about where the denominator can and can't be zero. Remember, you can't divide by zero! So, we need to keep that in the back of our minds as we go through the steps. We'll also need to consider cases to handle the potential sign changes that can happen when dealing with inequalities, especially when dealing with variables in the denominator. The goal is to isolate ${x}$ on one side of the inequality. We'll use a combination of algebraic manipulations, keeping in mind the rules of inequalities, to achieve this. Remember, the final solution will be expressed in interval notation, which is a concise way to represent the set of all values that satisfy the inequality. This will give us a clear view of the range of ${x}$ values that make the original statement true. So, now, let's see how this all comes together to solve ${\frac{1}{x+3}<2}$.

Step-by-Step Solution to the Inequality

Okay, guys, let's roll up our sleeves and solve the inequality ${\frac{1}{x+3}<2}$. Here's how we're going to break it down, step by step, to make sure we don't miss anything. First things first, we need to get rid of the fraction. A common (and often incorrect) approach is to multiply both sides by ${x+3}$. However, this can be problematic. If ${x+3}$ is negative, then we would need to flip the inequality sign, and if ${x+3}$ is positive, we do not. Since we do not know the sign of ${x+3}$, we need a different approach. Instead, let's rearrange the inequality so that one side is zero. We can do this by subtracting 2 from both sides of the inequality. This gives us ${\frac{1}{x+3} - 2 < 0}$. Next, we need to combine the terms on the left side into a single fraction. To do this, we need a common denominator, which in this case will be ${x+3}$. We can rewrite 2 as ${\frac{2(x+3)}{x+3}}$. So, our inequality becomes ${\frac{1}{x+3} - \frac{2(x+3)}{x+3} < 0}$. Now, we combine the fractions: ${\frac{1 - 2(x+3)}{x+3} < 0}$. Let's simplify the numerator: ${\frac{1 - 2x - 6}{x+3} < 0}$, which further simplifies to ${\frac{-2x - 5}{x+3} < 0}$. This is much easier to work with.

At this point, we need to find the critical values, which are the values of ${x}$ where the expression ${\frac{-2x - 5}{x+3}}$ equals zero or is undefined. This helps us to break down the number line into intervals. The expression is equal to zero when the numerator is zero. So, solve ${-2x - 5 = 0}$, which gives us ${x = -\frac{5}{2}}$. The expression is undefined when the denominator is zero. So, solve ${x+3=0}$, which gives us ${x=-3}$. Thus, the critical points are ${x = -\frac{5}{2}}$ and ${x = -3}$. These points divide the number line into three intervals: ${(-\infty, -\frac{5}{2})}$, ${(-\frac{5}{2}, -3)}$, and ${(-3, +\infty)}$. Now, we test a value from each of these intervals to see if it satisfies the inequality ${\frac{-2x - 5}{x+3} < 0}$. This helps us determine which intervals are part of our solution.

For the interval ${(-\infty, -\frac{5}{2})}$, let's test ${x=-3}$. Plugging this into ${\frac{-2x - 5}{x+3}}$, we get ${\frac{-2(-4) - 5}{-4+3} = \frac{3}{-1} = -3 < 0}$. The inequality holds, so this interval is part of our solution. For the interval ${(-\frac{5}{2}, -3)}$, let's test ${x = -\frac{7}{2}}$. Plugging this into ${\frac{-2x - 5}{x+3}}$, we get ${\frac{-2(-\frac{7}{2}) - 5}{-\frac{7}{2}+3} = \frac{2}{-\frac{1}{2}} = -4 < 0}$. The inequality does not hold. For the interval ${(-3, +\infty)}$, let's test ${x = 0}$. Plugging this into ${\frac{-2x - 5}{x+3}}$, we get ${\frac{-2(0) - 5}{0+3} = -\frac{5}{3} < 0}$. The inequality holds, so this interval is part of our solution. Finally, we exclude ${x=-3}$ because it makes the denominator zero (and the expression undefined), and we exclude ${x=-\frac{5}{2}}$ because it makes the expression equal to zero (and the original inequality is strict). Therefore, our solution consists of the intervals ${(-\infty, -\frac{5}{2})}$ and ${(-3, \infty)}$.

Expressing the Solution in Interval Notation

Alright, so after all that work, we've found the solution to our inequality. Now, let's express it in interval notation. Interval notation is a concise way to represent the range of values that satisfy an inequality. It's super useful because it's clear and easy to understand once you get the hang of it. From our previous steps, we found that the solution consists of two intervals: ${(-\infty, -\frac{5}{2})}$ and ${(-3, \infty)}$. In interval notation, we use parentheses ${()}$ to indicate that an endpoint is not included (meaning the inequality is strictly less than or greater than) and square brackets ${[ ]}$ to indicate that an endpoint is included (meaning the inequality is less than or equal to or greater than or equal to). Since our inequality is ${\frac{-2x - 5}{x+3} < 0}$, meaning the values are strictly less than zero, we'll use parentheses for all the endpoints. Because our solution has two separate intervals, we will write them using the union symbol, ${\cup}$. This symbol simply means