Hey guys! Today, we're diving deep into the fascinating world of inequalities, specifically focusing on how to find the solution set for the inequality -4(x+3) ≤ -2 - 2x and represent it on a number line. Inequalities might seem a bit daunting at first, but trust me, with a step-by-step approach, you'll be solving them like a pro in no time. So, buckle up and let's get started!
Understanding Inequalities
Before we jump into solving our specific inequality, let's take a moment to understand what inequalities are all about. Unlike equations, which have a single solution or a finite set of solutions, inequalities represent a range of values that satisfy a given condition. Think of it like this: instead of finding one specific number that makes an equation true, we're finding all the numbers that make an inequality true.
Inequalities use symbols like "less than" (<), "greater than" (>), "less than or equal to" (≤), and "greater than or equal to" (≥) to express relationships between values. For example, x > 5 means that x can be any number greater than 5, but not including 5 itself. On the other hand, x ≥ 5 means that x can be any number greater than or equal to 5, including 5.
The solution set for an inequality is the set of all values that make the inequality true. Our main goal here is to isolate the variable (in this case, 'x') on one side of the inequality to determine this solution set. And that's where the magic happens, guys!
Step-by-Step Solution for -4(x+3) ≤ -2 - 2x
Now, let's tackle our inequality: -4(x+3) ≤ -2 - 2x. We'll break it down into manageable steps to make sure everyone's on board. No one gets left behind!
Step 1: Distribute
Our first move is to get rid of the parentheses by distributing the -4 across the (x+3) term. Remember, this means multiplying -4 by both x and 3. So, we get:
-4 * x = -4x -4 * 3 = -12
This transforms our inequality into:
-4x - 12 ≤ -2 - 2x
See? We're already making progress! Distributing is a fundamental step in simplifying many algebraic expressions and inequalities.
Step 2: Combine Like Terms (Get x Terms on One Side)
Next, we want to gather all the terms with 'x' on one side of the inequality. A common practice is to move them to the side where the coefficient of 'x' will be positive. In our case, we have -4x on the left and -2x on the right. Adding 4x to both sides will keep things positive and neat. So, we add 4x to both sides:
-4x - 12 + 4x ≤ -2 - 2x + 4x
Simplifying this gives us:
-12 ≤ -2 + 2x
Awesome! We've successfully moved the 'x' terms to one side. Now, let's get those constant terms sorted.
Step 3: Isolate the Variable Term
Our next mission is to isolate the term with 'x' (which is 2x) by getting rid of the constant term on the same side. In this case, we have -2 on the right side. To eliminate it, we'll add 2 to both sides of the inequality:
-12 + 2 ≤ -2 + 2x + 2
Simplifying, we get:
-10 ≤ 2x
Looking good, guys! We're almost there. Just one more step to go.
Step 4: Solve for x
Finally, to get 'x' all by itself, we need to get rid of the coefficient 2. We do this by dividing both sides of the inequality by 2:
-10 / 2 ≤ 2x / 2
This gives us our solution:
-5 ≤ x
Or, we can rewrite it as:
x ≥ -5
This means that the solution set includes all values of x that are greater than or equal to -5. Boom! We've cracked the code.
Representing the Solution on a Number Line
Now that we've found our solution set, x ≥ -5, let's see how we can represent this visually on a number line. A number line is a straight line where numbers are placed at equal intervals, and it's a super helpful tool for visualizing inequalities.
Step 1: Draw the Number Line
First, we draw a horizontal line. Place zero (0) somewhere in the middle, and then mark off equal intervals to the left and right, representing positive and negative numbers. It doesn't have to be perfect, just clear and readable.
Step 2: Locate the Key Number
Our key number is -5, which is the boundary of our solution set. Find -5 on your number line and mark it.
Step 3: Use a Circle or a Bracket
This is where things get a little nuanced. Since our inequality includes "or equal to" (≥), we need to indicate that -5 itself is part of the solution set. We do this by using a closed circle (or a bracket) at -5. A closed circle means the number is included.
If our inequality was just x > -5 (without the "or equal to"), we would use an open circle at -5 to show that -5 is not included in the solution set.
Step 4: Shade the Solution Set
Now, we need to show all the values that satisfy x ≥ -5. This means all numbers greater than or equal to -5. On the number line, these numbers are to the right of -5. So, we shade the line to the right of -5, including the closed circle at -5. The shading indicates that all the numbers in that region are part of the solution set.
Key Takeaway: The direction of the shading tells you which values are included in the solution. If x is greater than a number, shade to the right. If x is less than a number, shade to the left.
Common Mistakes to Avoid
Alright, let's talk about some common pitfalls that people sometimes stumble into when solving inequalities. Knowing these mistakes will help you steer clear and ace those problems!
Mistake 1: Forgetting to Flip the Inequality Sign
This is a big one! When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, if you have -2x < 6, dividing both sides by -2 gives you x > -3 (notice how the "less than" sign became a "greater than" sign).
Mistake 2: Incorrectly Distributing
We touched on distribution earlier, but it's worth emphasizing. Make sure you distribute correctly, paying attention to signs. For instance, -3(x - 2) should be -3x + 6, not -3x - 6.
Mistake 3: Misinterpreting the Number Line Representation
Pay close attention to whether you should use an open circle or a closed circle (or a bracket). Remember, a closed circle (or bracket) means the number is included in the solution, while an open circle means it's not.
Mistake 4: Not Checking Your Solution
It's always a good idea to check your solution by plugging a value from your solution set back into the original inequality. If the inequality holds true, you're on the right track!
Practice Makes Perfect
The best way to master inequalities is to practice, practice, practice! Try solving a variety of inequalities, and don't be afraid to make mistakes – that's how we learn. Work through examples, check your answers, and ask for help when you need it.
Solving inequalities is a crucial skill in mathematics, and it's something you'll use in many different contexts. By understanding the fundamentals, avoiding common mistakes, and practicing regularly, you'll become an inequality-solving superstar!
So, there you have it, guys! A comprehensive guide to solving inequalities and representing the solution set on a number line. Remember, math can be fun, especially when you break it down step by step. Keep up the great work, and I'll catch you in the next one!