Solve (3/x) - 1 ≤ 1/(x+1): A Step-by-Step Guide

by Mei Lin 48 views

Hey guys! Let's dive into solving this inequality: (3/x) - 1 ≤ 1/(x+1). This is a fun one that combines fractions and inequalities, so we need to be careful with our steps. We'll break it down piece by piece to make sure we understand every move. Inequalities like this pop up all the time in calculus and other advanced math courses, so mastering them now is a great investment in your future math skills. We'll not only find the solution but also discuss why each step is crucial, paying special attention to those pesky denominators and how they affect the sign of our inequality. We'll also visualize the solution on a number line, which can be super helpful for understanding what our answer actually means. This isn't just about crunching numbers; it's about building a solid foundation for tackling more complex problems later on. We'll keep things conversational and explain everything in plain English, so don't worry if you're not a math whiz – we're all in this together! So, grab your pencils and paper, and let's get started! Remember, math is a journey, not a destination, and every problem we solve together makes us a little bit stronger. We’ll start by identifying the values of x that make the denominators zero, as these values will be excluded from our solution set. Then we will manipulate the inequality to bring all terms to one side, combine them into a single fraction, and analyze the sign of the resulting expression. We'll pay close attention to how the sign changes at the critical points (zeros of the numerator and denominator). Finally, we'll express our solution in interval notation, which is a concise way to represent the set of all x values that satisfy the inequality.

1. Finding the Critical Values

The first thing we need to do when tackling inequalities involving fractions is to identify any values of x that would make the denominators zero. Why? Because division by zero is a big no-no in mathematics! It makes the expression undefined. In our inequality, (3/x) - 1 ≤ 1/(x+1), we have two denominators: x and (x+1). So, let's find the values that make them zero:

  • x = 0: This is pretty straightforward. If x is 0, then the first fraction, 3/x, becomes undefined. So, x cannot be 0.
  • x + 1 = 0: To solve this, we subtract 1 from both sides, giving us x = -1. If x is -1, the second fraction, 1/(x+1), becomes undefined. So, x cannot be -1 either.

These values, 0 and -1, are super important. We call them critical values. They are the points where the expression on either side of the inequality might change its sign. Think of them as potential roadblocks on our number line. They divide the number line into different intervals, and within each interval, the expression will have a consistent sign (either positive or negative). These critical values will be crucial when we get to the sign analysis stage. They help us determine where the inequality holds true. By excluding these values from our final solution, we ensure that we're only considering the values of x for which the expression is properly defined. This step is often overlooked, but it's fundamental for solving rational inequalities correctly.

2. Rearranging the Inequality

Okay, now that we've identified our critical values, let's rearrange the inequality to make it easier to work with. Our goal here is to get everything on one side of the inequality sign, leaving zero on the other side. This will allow us to combine the terms into a single fraction and analyze its sign. Remember, we have: (3/x) - 1 ≤ 1/(x+1)

To get everything on the left side, we'll subtract 1/(x+1) from both sides. This gives us:

(3/x) - 1 - 1/(x+1) ≤ 0

Now, we need to combine these terms into a single fraction. To do that, we need a common denominator. The least common denominator (LCD) for x and (x+1) is simply their product: x(x+1). So, we'll rewrite each term with this denominator:

  • (3/x) becomes (3(x+1)) / (x(x+1))
  • -1 becomes -(x(x+1)) / (x(x+1))
  • -1/(x+1) becomes -(x) / (x(x+1))

Substituting these back into our inequality, we get:

(3(x+1) - x(x+1) - x) / (x(x+1)) ≤ 0

This step is all about algebraic manipulation. We're using the properties of fractions to combine terms, which is a fundamental skill in algebra. A common mistake is to skip steps or try to do too much at once, which can lead to errors. It's always a good idea to write out each step clearly to minimize the chance of making a mistake. Now that we have a single fraction, the next step is to simplify the numerator and analyze the sign of the resulting expression.

3. Simplifying the Numerator

Alright, let's simplify the numerator of our combined fraction. We have: (3(x+1) - x(x+1) - x) / (x(x+1)) ≤ 0

We need to expand and combine like terms in the numerator. Let's do it step by step:

  1. Distribute the 3: 3(x+1) = 3x + 3
  2. Distribute the -x: -x(x+1) = -x² - x

Now, substitute these back into the numerator:

3x + 3 - x² - x - x

Combine the like terms:

-x² + (3x - x - x) + 3 = -x² + x + 3

So, our inequality now looks like this:

(-x² + x + 3) / (x(x+1)) ≤ 0

The numerator is now a quadratic expression. Simplifying the numerator is a crucial step because it allows us to identify the zeros of the numerator, which are the values of x that make the numerator equal to zero. These zeros, along with the critical values we found earlier (0 and -1), will be the points where the sign of the entire expression might change. To find the zeros of the numerator, we'll need to solve the quadratic equation -x² + x + 3 = 0. We can use the quadratic formula for this, or try to factor the expression (though in this case, factoring isn't straightforward). Simplifying algebraic expressions is a fundamental skill in mathematics, and it's essential for solving a wide range of problems. By carefully expanding and combining like terms, we've transformed a complex expression into a simpler one that's easier to analyze.

4. Finding the Zeros of the Numerator

Now, let's find the zeros of the numerator, which is the quadratic expression -x² + x + 3. To do this, we need to solve the equation -x² + x + 3 = 0. Since this quadratic doesn't factor easily, we'll use the quadratic formula. Remember the quadratic formula? It's a handy tool for solving any quadratic equation of the form ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / (2a)

In our case, a = -1, b = 1, and c = 3. Let's plug these values into the formula:

x = (-1 ± √(1² - 4(-1)(3))) / (2(-1))

Simplify:

x = (-1 ± √(1 + 12)) / (-2) x = (-1 ± √13) / (-2)

So, we have two zeros:

  • x₁ = (-1 + √13) / (-2) ≈ -1.303
  • x₂ = (-1 - √13) / (-2) ≈ 2.303

These are the points where the numerator of our fraction becomes zero. Just like our critical values from the denominators, these zeros are crucial because they are potential points where the sign of the entire expression can change. Now we have a complete set of critical points: -1.303, -1, 0, and 2.303. These points will divide the number line into intervals, and we'll need to test each interval to see where our inequality holds true. Finding the roots of a quadratic equation is a fundamental skill in algebra, and the quadratic formula is a powerful tool for this. It's important to remember the formula and how to apply it correctly. We've now identified all the critical points that will help us solve the inequality.

5. Sign Analysis

Okay, this is where things get interesting! We've found our critical values and zeros: -1.303, -1, 0, and 2.303. These points divide the number line into five intervals: (-∞, -1.303), (-1.303, -1), (-1, 0), (0, 2.303), and (2.303, ∞). Now, we need to figure out the sign of our expression, (-x² + x + 3) / (x(x+1)), in each of these intervals. To do this, we'll pick a test value within each interval and plug it into the expression. If the result is negative or zero, the inequality is satisfied in that interval. If it's positive, the inequality is not satisfied.

Here’s how we'll do it:

  1. Interval (-∞, -1.303): Let's pick x = -2. Plugging into our expression: (-(−2)² + (−2) + 3) / (−2(−2+1)) = (-4 - 2 + 3) / (-2(-1)) = -3 / 2. This is negative, so the inequality is satisfied in this interval.
  2. Interval (-1.303, -1): Let's pick x = -1.1. Plugging into our expression: (-(-1.1)² + (-1.1) + 3) / (-1.1(-1.1+1)) ≈ 0.79 / 0.11. This is positive, so the inequality is not satisfied in this interval.
  3. Interval (-1, 0): Let's pick x = -0.5. Plugging into our expression: (-(-0.5)² + (-0.5) + 3) / (-0.5(-0.5+1)) = (2.25) / (-0.25). This is negative, so the inequality is satisfied in this interval.
  4. Interval (0, 2.303): Let's pick x = 1. Plugging into our expression: (-(1)² + (1) + 3) / (1(1+1)) = 3 / 2. This is positive, so the inequality is not satisfied in this interval.
  5. Interval (2.303, ∞): Let's pick x = 3. Plugging into our expression: (-(3)² + (3) + 3) / (3(3+1)) = (-3) / 12. This is negative, so the inequality is satisfied in this interval.

This sign analysis is the heart of solving inequalities like this. By systematically testing intervals, we can determine where the expression is positive, negative, or zero. It's important to be organized and careful with your calculations to avoid mistakes. A sign chart can be a helpful visual aid for keeping track of the signs in each interval. We're now one step closer to finding our final solution!

6. Writing the Solution in Interval Notation

Awesome! We've done the heavy lifting – we've found the critical values, simplified the expression, found the zeros, and performed the sign analysis. Now, it's time to write our solution in interval notation. Remember, we're looking for the intervals where our expression, (-x² + x + 3) / (x(x+1)), is less than or equal to zero. Based on our sign analysis, the inequality is satisfied in the following intervals:

  • (-∞, -1.303]
  • (-1, 0)
  • [2.303, ∞)

Let's break down why we use the brackets and parentheses:

  • Brackets [ ]: We use brackets to include the endpoints in the solution. In our case, we include -1.303 and 2.303 because the inequality is less than or equal to zero, so the zeros of the numerator are part of the solution.
  • Parentheses ( ): We use parentheses to exclude the endpoints from the solution. We exclude -1 and 0 because these values make the denominator zero, and the expression is undefined at these points.

So, the final solution in interval notation is:

(-∞, (-1 + √13) / (-2)] ∪ (-1, 0) ∪ [(-1 - √13) / (-2), ∞)

Or, approximately:

(-∞, -1.303] ∪ (-1, 0) ∪ [2.303, ∞)

This interval notation represents all the values of x that satisfy the original inequality. We've used the union symbol (∪) to combine the intervals into a single solution set. Writing the solution in interval notation is a concise and standard way to express the solution of an inequality. It's important to understand the meaning of the notation and how it relates to the sign analysis we performed. We've successfully navigated this complex inequality and arrived at our final answer!

Conclusion

Wow, guys, we made it! We successfully solved the inequality (3/x) - 1 ≤ 1/(x+1). We started by identifying the critical values, rearranged the inequality, simplified the numerator, found the zeros, performed a sign analysis, and finally, wrote the solution in interval notation. This problem demonstrates the importance of several key algebraic skills, including working with fractions, solving quadratic equations, and analyzing inequalities. Remember, the key to success in math is to break down complex problems into smaller, manageable steps. By carefully following each step and paying attention to detail, we can solve even the most challenging problems. Inequalities like this are fundamental in calculus and other advanced math topics, so mastering these skills now will set you up for success in the future. Keep practicing, and don't be afraid to ask questions. Math is a journey, and every problem you solve makes you stronger! Remember, always double-check your work and make sure your answer makes sense in the context of the problem. Great job, everyone!