Solving (5x + 2) / (4x - 1) ≥ 2: A Step-by-Step Guide With Graph And Interval Notation

by Mei Lin 87 views

Hey guys! Let's dive into solving the inequality (5x + 2) / (4x - 1) ≥ 2. Inequalities might seem tricky at first, but trust me, once you get the hang of it, they're pretty straightforward. We'll break it down step by step, graph the solution, and express it in interval notation. So, grab your pencils and let's get started!

Understanding Inequalities

Before we jump into the nitty-gritty, let's quickly recap what inequalities are all about. Unlike equations that have a single solution (or a few), inequalities deal with a range of values. Think of it like this: instead of finding the exact value of 'x' that makes an equation true, we're looking for all the values of 'x' that satisfy a certain condition, like being greater than, less than, greater than or equal to, or less than or equal to another value.

In this case, we're dealing with a rational inequality, which means we have a fraction with 'x' in both the numerator and the denominator. These types of inequalities require a bit more care because we need to consider the values of 'x' that make the denominator zero, as division by zero is a big no-no in the math world. These values are critical points that can change the sign of the expression, affecting the solution.

When you're tackling inequalities, it's super important to remember the golden rule: multiplying or dividing both sides by a negative number flips the inequality sign. This is a common pitfall, so keep it in mind! Another crucial aspect is to identify any values of 'x' that would make the denominator of a fraction equal to zero. These values are excluded from the solution set because they make the expression undefined. In our case, we'll need to keep a close eye on the value that makes 4x - 1 equal to zero.

Step-by-Step Solution

1. Move Everything to One Side

The first thing we need to do is get all the terms on one side of the inequality, leaving zero on the other side. This is similar to how we solve equations, but with the added complexity of the inequality sign. So, let's subtract 2 from both sides:

(5x + 2) / (4x - 1) - 2 ≥ 0

2. Find a Common Denominator

Now, to combine the terms, we need a common denominator. In this case, it's (4x - 1). We'll rewrite 2 as a fraction with this denominator:

(5x + 2) / (4x - 1) - 2 * (4x - 1) / (4x - 1) ≥ 0

This gives us:

(5x + 2 - 2(4x - 1)) / (4x - 1) ≥ 0

3. Simplify the Expression

Next, let's simplify the numerator by distributing the -2 and combining like terms:

(5x + 2 - 8x + 2) / (4x - 1) ≥ 0

This simplifies to:

(-3x + 4) / (4x - 1) ≥ 0

4. Find Critical Points

Now comes a crucial step: finding the critical points. These are the values of 'x' that make either the numerator or the denominator equal to zero. They are the potential turning points where the expression can change its sign.

  • Numerator: -3x + 4 = 0 => x = 4/3
  • Denominator: 4x - 1 = 0 => x = 1/4

So, our critical points are x = 1/4 and x = 4/3. Remember, x = 1/4 is particularly important because it makes the denominator zero, meaning it's not part of our solution set.

5. Create a Sign Chart

To determine the intervals where the inequality holds true, we'll create a sign chart. This chart helps us visualize how the expression (-3x + 4) / (4x - 1) changes its sign around the critical points. Draw a number line and mark the critical points 1/4 and 4/3 on it. These points divide the number line into three intervals:

  • (-∞, 1/4)
  • (1/4, 4/3)
  • (4/3, ∞)

Now, we'll pick a test value from each interval and plug it into our simplified expression (-3x + 4) / (4x - 1) to see if it's positive or negative.

  • Interval (-∞, 1/4): Let's pick x = 0 (-3(0) + 4) / (4(0) - 1) = 4 / -1 = -4 (Negative)
  • Interval (1/4, 4/3): Let's pick x = 1 (-3(1) + 4) / (4(1) - 1) = 1 / 3 (Positive)
  • Interval (4/3, ∞): Let's pick x = 2 (-3(2) + 4) / (4(2) - 1) = -2 / 7 (Negative)

6. Determine the Solution

We're looking for the intervals where (-3x + 4) / (4x - 1) ≥ 0, meaning where the expression is positive or equal to zero. From our sign chart, we see that this occurs in the interval (1/4, 4/3). Since we have a "greater than or equal to" sign, we include the value that makes the numerator zero (x = 4/3) but exclude the value that makes the denominator zero (x = 1/4).

Therefore, the solution is the interval (1/4, 4/3].

Graphing the Solution

Okay, so now that we've found the solution, let's visualize it on a number line. This will give us a clear picture of the range of values that satisfy our inequality. Here’s how we do it:

  1. Draw a number line: Grab a ruler (or just freehand it, no judgment here!) and draw a straight line. This is our canvas for the solution.
  2. Mark the critical points: Remember those critical points we found, 1/4 and 4/3? Let’s mark them on the number line. Since 1/4 is not included in the solution (because it makes the denominator zero), we'll use an open circle or parenthesis at this point. For 4/3, which is included in the solution, we'll use a closed circle or bracket.
  3. Shade the solution interval: We know our solution lies between 1/4 and 4/3, so shade the section of the number line between these two points. This shaded area represents all the values of 'x' that satisfy the inequality.
  4. Express the solution: You should have an open circle at 1/4 and a closed circle at 4/3, with the line segment in between shaded. This is a visual representation of our solution, clearly showing the range of 'x' values that make the inequality true. Visualizing solutions on a graph makes it much easier to understand the range of possible values and reinforces the concept of inequalities representing a range rather than a single point. For those who learn visually, this step is super helpful in solidifying the solution. It's like a quick check to make sure the interval notation we'll write next makes sense in terms of the graph. If your graph doesn’t match your interval notation, it's a sign to go back and double-check your work.

Interval Notation

Interval notation is a concise way to represent the solution set of an inequality. It uses parentheses and brackets to indicate whether the endpoints are included or excluded. Let's translate our solution into interval notation:

  • A parenthesis '(' indicates that the endpoint is not included (open interval).
  • A bracket '[' indicates that the endpoint is included (closed interval).
  • We use '∞' (infinity) and '-∞' (negative infinity) to represent intervals that extend indefinitely.

In our case, the solution includes all values between 1/4 and 4/3, including 4/3 but excluding 1/4. Therefore, the interval notation for our solution is:

(1/4, 4/3]

See how the parenthesis is next to 1/4, showing it's not included, and the bracket is next to 4/3, showing it is included? Interval notation is like a secret code that mathematicians use to quickly and accurately describe solution sets. It’s efficient and gets straight to the point. Once you get the hang of reading and writing it, you’ll find it’s a really useful tool.

Common Mistakes to Avoid

Inequalities can be a bit tricky, so it's helpful to be aware of some common mistakes people make. Avoiding these pitfalls can save you a lot of headaches and ensure you arrive at the correct solution. Let's highlight a few key areas where errors often pop up:

  1. Forgetting to flip the sign: Remember that golden rule we talked about? If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is a crucial step, and forgetting it will lead to the wrong answer. Think of it as a reflex – negative number, flip the sign!
  2. Ignoring critical points: Critical points are the values that make either the numerator or the denominator of a rational expression equal to zero. These points are like road signs, telling you where the expression might change its behavior. Ignoring them means you're missing vital information about the solution.
  3. Including values that make the denominator zero: We've emphasized this before, but it's worth repeating: a denominator of zero is a big no-no in math. Values that make the denominator zero are not part of the solution set and should be excluded. Using open circles or parentheses in your graph and interval notation is how you signal this exclusion.
  4. Incorrectly interpreting the sign chart: The sign chart is your friend, but it only works if you interpret it correctly. Make sure you're picking test values within each interval and plugging them into the simplified expression. A small arithmetic error here can throw off your entire solution. Double-check your calculations to avoid this.
  5. Mixing up parentheses and brackets: Parentheses and brackets have very specific meanings in interval notation. Parentheses mean the endpoint is not included, while brackets mean it is. Using the wrong one can completely change the solution set you're describing. Make sure you understand the difference and use them appropriately based on whether the critical points are included or excluded.
  6. Not simplifying the expression correctly: Before you start finding critical points and building sign charts, make sure your expression is simplified as much as possible. Errors in simplification can lead to incorrect critical points and a flawed solution. Take your time and double-check your algebraic manipulations.

By being aware of these common mistakes, you can approach inequalities with confidence and increase your chances of getting the right answer. Remember, practice makes perfect, so keep working through examples and challenging yourself!

Conclusion

And there you have it! We've successfully solved the inequality (5x + 2) / (4x - 1) ≥ 2, graphed the solution, and expressed it in interval notation. Remember, the key is to break down the problem into manageable steps, find those critical points, use a sign chart, and pay attention to the details. Inequalities might seem daunting at first, but with practice, you'll be solving them like a pro. Keep up the great work, guys!