Solving And Graphing Inequalities A Step By Step Guide

Hey guys! Let's dive into the world of inequalities and how to represent their solutions graphically. We've got a fun problem to tackle today that involves solving an inequality and then figuring out which graph correctly illustrates the solution. So, grab your thinking caps, and let's get started!

Understanding the Inequality

Our main task is to solve the inequality: -2.4(x - 6) ≥ 52.8. Inequalities like this one are mathematical statements that compare two expressions using symbols like greater than (>) or greater than or equal to (≥), less than (<) or less than or equal to (≤), or not equal to (≠). Unlike equations, which have one specific solution (or a few), inequalities often have a range of solutions. Think of it like finding all the numbers that make the statement true, not just one.

To solve this, our main goal is to isolate the variable 'x' on one side of the inequality. This will tell us the range of values that 'x' can take to satisfy the inequality. It's a bit like solving an equation, but there's one crucial difference we'll need to keep in mind.

The Golden Rule of Inequalities

Before we dive into the steps, let's quickly review the golden rule of inequalities: when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is super important because it's a common mistake people make. Imagine it like this: if you have 5 > 3 (which is true), but you multiply both sides by -1, you get -5 > -3, which is not true. To make it true, you need to flip the sign: -5 < -3. Got it? Great!

Step-by-Step Solution

Okay, now let's break down the steps to solve our inequality:

1. Distribute: First, we need to get rid of those parentheses. Distribute the -2.4 across the terms inside the parentheses:

   $$-2.4 * x + (-2.4 * -6) ≥ 52.8$$

   $$-2.4x + 14.4 ≥ 52.8$$

2. Isolate the variable term: Next, we want to get the term with 'x' by itself on one side. To do this, we'll subtract 14.4 from both sides of the inequality:

   $$-2.4x + 14.4 - 14.4 ≥ 52.8 - 14.4$$

   $$-2.4x ≥ 38.4$$

3. Solve for x: Now, we need to get 'x' completely alone. This means dividing both sides by -2.4. But remember that golden rule! We're dividing by a negative number, so we need to flip the inequality sign:

   $$\frac{-2.4x}{-2.4} ≤ \frac{38.4}{-2.4}$$

   $$x ≤ -16$$

So, our solution is x ≤ -16. This means that any value of 'x' that is less than or equal to -16 will satisfy the original inequality.

Graphing the Solution

Now that we've solved for 'x', let's talk about how to represent this solution graphically. When we're dealing with inequalities, we usually use a number line to visualize the range of possible solutions.

Understanding Number Lines

A number line is simply a line that represents all real numbers. Zero is in the middle, positive numbers go to the right, and negative numbers go to the left. When graphing inequalities, we use a few key elements:

• Circles or Brackets: We use a circle or a bracket to indicate the endpoint of the solution. A closed circle (or a square bracket) means that the endpoint is included in the solution (like with ≤ or ≥), while an open circle (or a parenthesis) means the endpoint is not included (like with < or >).
• Shading: We shade the portion of the number line that represents the range of solutions. If 'x' is less than a number, we shade to the left. If 'x' is greater than a number, we shade to the right.

Graphing x ≤ -16

Okay, let's graph our solution, x ≤ -16. Here's how we'll do it:

1. Find -16 on the number line: Locate -16 on your imaginary number line.
2. Draw a closed circle (or a square bracket) at -16: Since our inequality is 'less than or equal to', we use a closed circle (or a square bracket) to show that -16 is part of the solution.
3. Shade to the left: Since 'x' is less than -16, we shade everything to the left of -16. This indicates that all numbers less than -16 are also solutions to the inequality.

So, the graph of x ≤ -16 will have a closed circle (or a square bracket) at -16 and shading extending to the left.

Identifying the Correct Graph

Now, let's get back to our original question: Which graph represents the solution to the inequality -2.4(x - 6) ≥ 52.8? We've already done the hard work! We know that the solution is x ≤ -16, and we know what that looks like on a number line.

All you need to do is carefully examine the graphs provided (A, B, C, and D) and look for the one that has a closed circle (or a square bracket) at -16 and shading to the left. Whichever graph matches that description is the correct answer!

Common Mistakes to Avoid

Before we wrap up, let's quickly touch on some common mistakes people make when solving and graphing inequalities:

• Forgetting to flip the sign: This is the biggest one! Always remember to flip the inequality sign when multiplying or dividing by a negative number.
• Using the wrong type of circle/bracket: Make sure you're using a closed circle (or square bracket) for ≤ and ≥, and an open circle (or parenthesis) for < and >.
• Shading in the wrong direction: Double-check whether you need to shade to the left (for less than) or to the right (for greater than).

By keeping these pitfalls in mind, you'll be well on your way to mastering inequalities!

Conclusion

And that's it, guys! We've successfully solved the inequality -2.4(x - 6) ≥ 52.8 and learned how to represent its solution graphically. Remember, the key is to isolate the variable, pay attention to the golden rule of flipping the sign, and carefully interpret the graph. With a little practice, you'll become inequality-solving pros in no time! Keep up the great work, and I'll see you in the next math adventure!