Solving Inequality: Number Line Representation

Let's dive into the world of inequalities and number lines! This article will guide you through solving the inequality $-4(x+3) \leq -2-2x$ and accurately representing its solution set on a number line. So, grab your math hats, and let's get started!

Understanding Inequalities

Before we jump into the problem, let's quickly recap what inequalities are. Unlike equations that show equality between two expressions, inequalities show a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. These relationships are represented by the symbols >, <, $\geq$, and $\leq$.

Why are inequalities important? Well, in the real world, things aren't always equal. Think about setting a budget (you want your spending to be less than or equal to your income) or planning a road trip (you need to cover a distance greater than a certain amount). Inequalities help us model and solve these types of situations.

In our case, we are dealing with the inequality $-4(x+3) \leq -2-2x$. This tells us that the expression on the left side, $-4(x+3)$, is less than or equal to the expression on the right side, $-2-2x$. Our goal is to find all the values of 'x' that make this statement true. These values form the solution set, which we will then represent on a number line.

Solving the Inequality $-4(x+3) \leq -2-2x$

Now, let's get our hands dirty and solve this inequality step-by-step. Don't worry, it's not as scary as it looks! We'll use the same principles we use for solving equations, with one crucial difference that we'll highlight later.

Step 1: Distribute

Our first step is to simplify both sides of the inequality by distributing the -4 on the left side:

$-4(x+3) \leq -2-2x$ becomes $-4x - 12 \leq -2 - 2x$

Think of the distributive property as sharing the love (or in this case, the -4) with each term inside the parentheses. We multiply -4 by both 'x' and +3.

Step 2: Combine Like Terms

Next, we want to gather all the 'x' terms on one side of the inequality and the constant terms (the numbers) on the other side. To do this, we can add 4x to both sides:

$-4x - 12 + 4x \leq -2 - 2x + 4x$ simplifies to $-12 \leq -2 + 2x$

Now, let's add 2 to both sides to isolate the 'x' term further:

$-12 + 2 \leq -2 + 2x + 2$ simplifies to $-10 \leq 2x$

Step 3: Isolate the Variable

Our final step in solving for 'x' is to divide both sides of the inequality by 2:

$\frac{-10}{2} \leq \frac{2x}{2}$ simplifies to $-5 \leq x$

The crucial difference: When we multiply or divide both sides of an inequality by a negative number, we must flip the direction of the inequality sign. However, since we divided by a positive 2 here, we don't need to flip the sign.

Interpreting the Solution

So, what does $-5 \leq x$ actually mean? It means that 'x' can be any number that is greater than or equal to -5. In other words, -5 is the smallest possible value for 'x', and anything larger than -5 also works. You can also read this as "x is greater than or equal to -5".

Representing the Solution on a Number Line

Now comes the fun part: visualizing our solution on a number line! A number line is a visual representation of all real numbers, extending infinitely in both positive and negative directions. It's the perfect tool for showing the solution set of an inequality.

Step 1: Draw the Number Line

Start by drawing a horizontal line. Mark zero in the middle, and then mark some numbers to the left (negative) and right (positive) of zero. Make sure to include -5 on your number line, as it's a key value in our solution.

Step 2: Place a Dot or Circle

At the number -5, we need to place either a closed circle (also called a dot) or an open circle. This is where the "or equal to" part of our inequality comes into play.

• Closed Circle (Dot): We use a closed circle when the solution includes the number itself. This happens when we have $\leq$ (less than or equal to) or $\geq$ (greater than or equal to). In our case, we have $-5 \leq x$, which means -5 is part of the solution, so we use a closed circle.
• Open Circle: We use an open circle when the solution does not include the number itself. This happens when we have < (less than) or > (greater than).

So, on our number line, we'll place a solid dot at -5.

Step 3: Shade the Solution

Now we need to show all the other numbers that are part of the solution. Our inequality tells us that 'x' is greater than or equal to -5. This means we need to shade the portion of the number line to the right of -5, as all the numbers to the right of -5 are greater than it. Use a bold line or shading to clearly indicate this.

Putting it all Together

Your final number line should have a solid dot at -5 and a shaded line extending to the right, indicating all the numbers greater than -5. This visual representation perfectly captures the solution set of the inequality $-4(x+3) \leq -2-2x$.

Examples of Number Line Representation

To solidify your understanding, let's look at a couple more examples:

Example 1: x > 2

• We place an open circle at 2 because 2 is not included in the solution (x is strictly greater than 2).
• We shade the number line to the right of 2, representing all numbers greater than 2.

Example 2: x < -1

• We place an open circle at -1 because -1 is not included in the solution (x is strictly less than -1).
• We shade the number line to the left of -1, representing all numbers less than -1.

Example 3: x $\geq$ 3

• We place a closed circle at 3 because 3 is included in the solution (x is greater than or equal to 3).
• We shade the number line to the right of 3, representing all numbers greater than or equal to 3.

Example 4: x $\leq$ 0

• We place a closed circle at 0 because 0 is included in the solution (x is less than or equal to 0).
• We shade the number line to the left of 0, representing all numbers less than or equal to 0.

Common Mistakes to Avoid

Inequalities can be a bit tricky, so let's highlight some common mistakes to watch out for:

1. Forgetting to Flip the Inequality Sign: Remember, if you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is a very common mistake, so double-check your work!
2. Using the Wrong Circle: Make sure you use a closed circle (dot) for $\leq$ and $\geq$ and an open circle for < and >. The circle indicates whether the number itself is included in the solution or not.
3. Shading in the Wrong Direction: Always double-check which direction to shade. If 'x' is greater than a number, shade to the right. If 'x' is less than a number, shade to the left. Reading the inequality carefully will help you avoid this mistake.
4. Misinterpreting the Solution: Take a moment to understand what your solution actually means in the context of the original problem. This will help you avoid making errors in your final answer.

Real-World Applications

Inequalities aren't just abstract math concepts; they have tons of real-world applications. Here are a few examples:

• Budgeting: As mentioned earlier, inequalities are essential for budgeting. You might use an inequality to represent how much you can spend each month while staying within your income.
• Speed Limits: Speed limits are a classic example of inequalities. The speed you drive must be less than or equal to the posted speed limit.
• Age Restrictions: Many activities have age restrictions that can be expressed as inequalities. For example, you must be greater than or equal to 16 years old to get a driver's license in most places.
• Temperature Ranges: Think about the thermostat in your home. You might set a temperature range that you want to maintain, which can be expressed using inequalities.
• Inventory Management: Businesses use inequalities to manage their inventory levels. They need to make sure they have enough products in stock to meet demand but not so much that they have excess inventory.

By understanding inequalities, you're gaining a valuable tool for problem-solving in many different areas of life.

Conclusion

There you have it! You've learned how to solve the inequality $-4(x+3) \leq -2-2x$ and represent its solution set on a number line. Remember the key steps: distribute, combine like terms, isolate the variable, and don't forget to flip the sign if you multiply or divide by a negative number. Understanding number line representations not only helps in math class but also provides a visual tool for interpreting inequalities in real-world scenarios. Keep practicing, and you'll become a number line pro in no time!

So next time you see an inequality, don't fret! Just remember the steps, visualize the solution on a number line, and you'll be well on your way to solving it. You've got this, guys! Now go forth and conquer those inequalities!