Solving Inequalities Step By Step 5 ≤ 4x - 3 < 9

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Hey everyone! Let's dive into solving a common type of inequality problem today. We've got a double inequality here, which might look a little intimidating at first, but I promise it's totally manageable. Our goal is to isolate 'x' in the middle, so we know exactly what values 'x' can be.

Understanding Double Inequalities

First, let's break down what this inequality, 5 ≤ 4x - 3 < 9, actually means. It's like saying we have two inequalities happening at the same time. We're saying that '4x - 3' is both greater than or equal to 5 AND less than 9. Think of it as '4x - 3' being trapped between 5 and 9. To solve for x, we need to get 'x' all by itself in the middle. We do this by performing the same operations on all three parts of the inequality – the left side, the middle, and the right side. This ensures we maintain the balance of the inequality.

Now, before we jump into the steps, let’s quickly refresh why we do the same thing to all parts. Imagine a balance scale. If you add or subtract weight from one side, you need to do the same on the other to keep it balanced. Inequalities are similar; whatever we do, we need to do it to all parts to maintain the relationship. Ignoring this principle can lead to incorrect solutions. Understanding this fundamental concept is crucial for accurately solving inequalities. It’s not just about following steps; it’s about understanding why those steps work. This approach will help you tackle more complex problems with confidence.

Step-by-Step Solution

Step 1: Isolate the Term with 'x'

Our first mission is to get the term with 'x' (which is '4x' in this case) by itself. Notice that we have a '- 3' hanging out with the '4x'. To get rid of it, we need to do the opposite operation: addition. We're going to add 3 to all three parts of the inequality. So, 5 ≤ 4x - 3 < 9 becomes 5 + 3 ≤ 4x - 3 + 3 < 9 + 3, which simplifies to 8 ≤ 4x < 12. See how we added 3 to the 5, the '4x - 3', and the 9? That's the key to keeping the inequality balanced.

Step 2: Isolate 'x'

Now we've got '4x' in the middle, but we want just 'x'. The '4' is multiplying the 'x', so to undo that, we need to divide. We'll divide all three parts of the inequality by 4. Remember, dividing by a positive number doesn't flip the inequality signs (we'll talk about negative numbers later). So, 8 ≤ 4x < 12 becomes 8 / 4 ≤ 4x / 4 < 12 / 4, which simplifies to 2 ≤ x < 3. And there you have it! We've isolated 'x'.

Step 3: Interpreting the Solution

So, what does 2 ≤ x < 3 actually mean? It means that 'x' can be any number that is greater than or equal to 2, but strictly less than 3. It's important to pay attention to the 'equal to' part. The '≤' means 'less than or equal to', so 2 is included in our solution. But the '<' means 'strictly less than', so 3 is not included. We can visualize this on a number line. We'd use a closed circle (or a square bracket) at 2 to show it's included, and an open circle (or a parenthesis) at 3 to show it's not included. This is a critical step, as it ensures you understand the range of values that satisfy the original inequality.

Representing the Solution

We can represent our solution in a few different ways:

  • Inequality Notation: 2 ≤ x < 3 (This is what we already have)
  • Number Line: (Imagine a number line with a closed circle at 2 and an open circle at 3, with the line shaded between them).
  • Interval Notation: [2, 3) (The square bracket indicates inclusion, and the parenthesis indicates exclusion).

Interval notation is a compact and common way to represent solutions to inequalities. The square bracket '[ ]' means the endpoint is included, while the parenthesis '( )' means the endpoint is not included. So, '[2, 3)' means all numbers from 2 up to (but not including) 3. Getting comfortable with these different notations will help you in more advanced math courses.

Special Cases and Common Mistakes

Let's talk about a couple of things that can trip people up when solving inequalities:

Dividing by a Negative Number

This is a big one: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, if we had -2x < 6, dividing by -2 would give us x > -3 (notice the sign flip!). This is because multiplying or dividing by a negative number reverses the order of the number line. For instance, -2 is less than -1, but if you multiply both by -1, you get 2 and 1, and now 2 is greater than 1. Forgetting to flip the sign is a very common mistake, so always double-check when you're working with negative numbers.

No Solution or All Real Numbers

Sometimes, when you solve an inequality, you might end up with a statement that's always false (like 5 < 2) or always true (like 0 < 5). If you get a false statement, it means there's no solution to the inequality. There's no value of 'x' that will make the original inequality true. If you get a true statement, it means all real numbers are solutions. Any value of 'x' will work. Recognizing these cases is essential for a complete understanding of inequality solutions.

Practice Makes Perfect

The best way to get comfortable with solving inequalities is to practice! Try some similar problems on your own. Remember to follow the steps carefully, and pay attention to those special cases. If you get stuck, don't be afraid to ask for help. Here's a practice problem you can try:

Solve for x: -3 < 2x + 1 ≤ 7

Work through it, and see if you can get the correct answer. You can check your solution by plugging it back into the original inequality to make sure it holds true.

Conclusion

Solving inequalities, especially double inequalities, might seem tricky at first, but with a clear understanding of the steps and a little practice, you'll be solving them like a pro in no time! Remember the key: perform the same operations on all parts to maintain balance, and don't forget to flip the sign when multiplying or dividing by a negative number. Keep practicing, and you'll master this skill in no time!