Solving Compound Inequalities: A Step-by-Step Guide

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Hey guys! Today, we're diving into the world of compound inequalities. Specifically, we're going to tackle an inequality that looks like this: −5\[HTMLREMOVED]6x+1<19-5 \[HTML_REMOVED] 6x + 1 < 19. Don't worry, it's not as scary as it looks! We'll break it down step by step, so you'll be solving these like a pro in no time. Our goal is to isolate 'x' in the middle, and then represent our solution both as an inequality and on a graph.

Understanding Compound Inequalities

So, what exactly is a compound inequality? Simply put, it's two or more inequalities joined together. In our case, we have −5\[HTMLREMOVED]6x+1-5 \[HTML_REMOVED] 6x + 1 and 6x+1<196x + 1 < 19 combined into one statement. This means that 6x+16x + 1 must be greater than or equal to -5 and less than 19. When you see a compound inequality like this, think of it as a set of conditions that 'x' must satisfy. Understanding this basic concept is crucial before we start manipulating the inequality to find the solution. We're essentially looking for a range of values for 'x' that make the entire statement true. This range will be bounded by two values, and 'x' will lie somewhere between them. In some cases, it might include one or both of the boundary values, depending on whether the inequality signs are strict (less than or greater than) or inclusive (less than or equal to, or greater than or equal to). Keep this in mind as we proceed, and you'll find that solving compound inequalities becomes much more intuitive.

Step 1: Isolate the Variable Term

Our first mission is to get the term with 'x' (in this case, 6x6x) all by itself in the middle. To do this, we need to get rid of that '+ 1'. Remember, whatever we do to one part of the inequality, we have to do to all parts to keep things balanced. So, we're going to subtract 1 from all three sections:

−5−1≤6x+1−1<19−1-5 - 1 \leq 6x + 1 - 1 < 19 - 1

This simplifies to:

−6≤6x<18-6 \leq 6x < 18

Alright, we're making progress! Subtracting 1 from each part of the inequality maintains the balance and helps us isolate the term with 'x'. This is a fundamental step in solving any inequality, whether it's a simple one or a more complex compound inequality like this one. Think of it as peeling away the layers to reveal the 'x' term underneath. Each operation we perform brings us closer to our goal of having 'x' all alone in the middle, so we can clearly see the range of values that satisfy the original inequality. Now that we've successfully subtracted 1, we're ready to move on to the next step and further isolate 'x'.

Step 2: Solve for x

Now we have −6≤6x<18-6 \leq 6x < 18. To finally isolate 'x', we need to get rid of the '6' that's multiplying it. How do we do that? You guessed it – we divide! We're going to divide all three parts of the inequality by 6:

−66≤6x6<186\frac{-6}{6} \leq \frac{6x}{6} < \frac{18}{6}

This gives us:

−1≤x<3-1 \leq x < 3

Boom! We've solved for 'x'! This means that 'x' can be any number greater than or equal to -1, but it has to be less than 3. Dividing each part of the inequality by the same positive number (in this case, 6) preserves the direction of the inequality signs. This is a crucial point to remember! If we were dividing by a negative number, we'd have to flip the inequality signs. But since we're dividing by a positive number, we can simply perform the division and arrive at our solution: −1≤x<3-1 \leq x < 3. This inequality tells us the range of values that 'x' can take to satisfy the original compound inequality. It's a clear and concise way to express the solution, and it's also the foundation for representing the solution graphically.

Step 3: Graph the Solution Set

Okay, so we know that −1≤x<3-1 \leq x < 3. How do we show this on a number line? Here's the breakdown:

  1. Draw a number line: Just a straight line with numbers marked on it. Make sure to include -1 and 3.
  2. Closed circle at -1: Since xx can be equal to -1 (because of the ≤\leq sign), we use a closed circle (or a filled-in dot) at -1 to show that -1 is included in the solution.
  3. Open circle at 3: Since xx has to be less than 3 (because of the << sign), we use an open circle at 3 to show that 3 is not included in the solution.
  4. Shade the line between -1 and 3: This shows that all the numbers between -1 and 3 (including -1, but not including 3) are part of the solution.

So, on your number line, you'll have a solid dot at -1, an open circle at 3, and the line shaded in between them. Graphing the solution set provides a visual representation of the range of values that satisfy the inequality. The closed circle at -1 indicates that -1 is included in the solution, while the open circle at 3 indicates that 3 is not included. The shaded line between -1 and 3 represents all the numbers that fall within the solution set. This visual representation can be particularly helpful for understanding the concept of inequalities and for quickly identifying the values that satisfy the given conditions. It's a powerful tool for both solving and interpreting inequalities.

Step 4: Express the Solution as an Inequality

Guess what? We already did this! The inequality −1≤x<3-1 \leq x < 3 is the solution. It tells us everything we need to know: 'x' is greater than or equal to -1 and less than 3. Sometimes, the question might specifically ask for interval notation, in which case the answer would be [−1,3)[-1, 3).

Expressing the solution as an inequality is a concise and precise way to communicate the range of values that satisfy the original problem. In this case, the inequality −1≤x<3-1 \leq x < 3 clearly defines the boundaries of the solution set and indicates whether the endpoints are included or excluded. This form of the solution is easy to understand and can be directly used for further calculations or analysis. It's a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. Understanding how to express solutions as inequalities is crucial for developing a solid foundation in mathematics.

Let's Recap

To solve the compound inequality −5≤6x+1<19-5 \leq 6x + 1 < 19, we:

  1. Subtracted 1 from all parts: −6≤6x<18-6 \leq 6x < 18
  2. Divided all parts by 6: −1≤x<3-1 \leq x < 3
  3. Graph the solution set on a number line with a closed circle at -1, an open circle at 3, and shading in between.
  4. Expressed the solution as the inequality: −1≤x<3-1 \leq x < 3

And that's it! You've successfully solved a compound inequality. Keep practicing, and you'll become a master in no time. Remember the key steps: isolate the variable term, solve for the variable, graph the solution set, and express the solution as an inequality. With these skills, you'll be well-equipped to tackle a wide range of mathematical problems. Now go out there and conquer those inequalities! Good luck, and have fun!