Graphing Solutions On A Number Line 6 - 2(x - 3) > 3x - 3

Hey guys! Today, we're diving into a fun math problem where we need to graph the solutions of an inequality on a number line. The inequality we're tackling is:

$6 - 2(x - 3) > 3x - 3$

Buckle up, because we're going to break this down step by step, making it super easy to understand. We'll cover everything from simplifying the inequality to plotting our solution on the number line. So, let's get started!

Understanding Inequalities

Before we jump into solving our specific problem, let's quickly recap what inequalities are all about. Inequalities, unlike equations, don't have a single solution. Instead, they define a range of values that make the statement true. Think of it like this: instead of finding one magic number, we're finding a whole bunch of numbers that work!

The symbols we use in inequalities are:

• (greater than)

• < (less than)
• ≥ (greater than or equal to)
• ≤ (less than or equal to)

These symbols tell us how the expressions on either side of the inequality relate to each other. For example, $x > 5$ means that x can be any number bigger than 5, but not 5 itself. If it was $x ≥ 5$, then x could be 5 or any number greater than 5.

When we graph inequalities on a number line, we use open circles for > and < (because the endpoint isn't included) and closed circles for ≥ and ≤ (because the endpoint is included). This visual representation helps us see all the possible solutions at a glance. It's like creating a map of all the numbers that make our inequality happy!

Step-by-Step Solution

Okay, let's get our hands dirty and solve the inequality $6 - 2(x - 3) > 3x - 3$. We'll take it one step at a time, so you can follow along easily.

1. Distribute

The first thing we need to do is get rid of those parentheses. We do this by distributing the -2 across the terms inside the parenthesis:

$6 - 2(x - 3) > 3x - 3$ becomes $6 - 2x + 6 > 3x - 3$

Remember, when you distribute a negative number, you need to pay close attention to the signs. $-2 * x$ is $-2x$, and $-2 * -3$ is +6. It’s a common spot for mistakes, so double-check your work!

2. Combine Like Terms

Now, let's simplify each side of the inequality by combining like terms. On the left side, we can combine the constants 6 and 6:

$6 - 2x + 6 > 3x - 3$ simplifies to $-2x + 12 > 3x - 3$

This step makes our inequality look much cleaner and easier to work with. It's like decluttering your workspace before tackling a big project!

3. Move Variables to One Side

Next, we want to get all the terms with x on one side of the inequality. A good strategy is to move the smaller x term to the other side to avoid dealing with negative coefficients. In this case, we'll add 2x to both sides:

$-2x + 12 > 3x - 3$ becomes $12 > 5x - 3$

Adding the same thing to both sides keeps the inequality balanced, just like in an equation. We're essentially shifting things around without changing the fundamental relationship.

4. Move Constants to the Other Side

Now, let's isolate the x term by moving the constant term (-3) to the other side. We do this by adding 3 to both sides:

$12 > 5x - 3$ becomes $15 > 5x$

We're getting closer! Just a little bit more algebra magic, and we'll have our solution.

5. Isolate the Variable

Finally, we need to get x all by itself. To do this, we'll divide both sides of the inequality by 5:

$15 > 5x$ becomes $3 > x$

Or, we can rewrite this as $x < 3$. Remember, $3 > x$ means the same thing as $x < 3$. It just says that x is less than 3.

6. Understanding the Solution

Our solution, $x < 3$, tells us that any number less than 3 will satisfy the original inequality. This isn't just one number; it's a whole range of numbers! Think of it as an infinite set of solutions.

For example, 2, 0, -1, -100, and even 2.999 are all solutions because they are less than 3. The number 3 itself is not a solution because our inequality is strictly "less than," not "less than or equal to."

Graphing the Solution on the Number Line

Now comes the visual part! Graphing our solution on the number line helps us see all the possible values of x that make our inequality true. Here's how we do it:

1. Draw a Number Line

First, draw a straight line and mark some numbers on it. You don't need to include every single number; just enough to give you a good sense of the scale. Make sure to include the key number from our solution, which is 3.

2. Place an Open Circle

Since our solution is $x < 3$, we use an open circle at 3. An open circle means that 3 is not included in the solution set. It's like saying, "We're getting really close to 3, but we're not quite there."

If our inequality was $x ≤ 3$, we would use a closed circle instead, indicating that 3 is included.

3. Shade the Line

Because x can be any number less than 3, we shade the number line to the left of the open circle. This shaded region represents all the possible solutions to our inequality. It's like coloring in all the numbers that make our inequality happy!

4. Add an Arrow

Finally, we add an arrow at the end of the shaded line to indicate that the solutions continue infinitely in that direction. This shows that there's no lower limit to the numbers that satisfy our inequality.

Putting It All Together

So, to graph the solution $x < 3$ on a number line, you'll have:

• A number line with some numbers marked on it.
• An open circle at 3.
• The line shaded to the left of 3.
• An arrow pointing to the left.

That's it! You've successfully graphed the solution to an inequality on a number line. Give yourself a pat on the back!

Common Mistakes to Avoid

Solving inequalities can be tricky, and it's easy to make small mistakes along the way. Here are a few common pitfalls to watch out for:

1. Forgetting to Distribute Negatives

When distributing a negative number, make sure you multiply it correctly with every term inside the parentheses. A missed negative sign can throw off your entire solution.

2. Not Flipping the Inequality Sign

Remember, when you multiply or divide both sides of an inequality by a negative number, you need to flip the inequality sign. For example, if you have $-2x < 6$, dividing by -2 gives you $x > -3$. Forgetting to flip the sign is a classic mistake.

3. Using the Wrong Type of Circle

Make sure you use the correct type of circle when graphing your solution. Open circles are for > and <, while closed circles are for ≥ and ≤. Using the wrong circle will give an inaccurate representation of your solution.

4. Shading in the Wrong Direction

Double-check that you're shading the number line in the correct direction. If your solution is $x < 3$, you should shade to the left of 3. If it's $x > 3$, you should shade to the right.

5. Not Checking Your Solution

It's always a good idea to check your solution by plugging in a value from your shaded region into the original inequality. If it works, you're on the right track! If not, go back and look for any mistakes.

Real-World Applications

Inequalities aren't just abstract math concepts; they show up in real-world situations all the time! Here are a few examples:

1. Budgeting

Imagine you have a budget of $50 for groceries. You can express this as an inequality: $spending ≤ $50. This means your total spending must be less than or equal to $50. Inequalities help you stay within your financial limits.

2. Speed Limits

Speed limits on roads are another example of inequalities. If the speed limit is 65 mph, this means your speed must be less than or equal to 65 mph. Going faster could get you a ticket!

3. Age Restrictions

Many activities have age restrictions that can be expressed as inequalities. For example, you might need to be at least 18 years old to vote. This can be written as $age ≥ 18$.

4. Temperature Ranges

Weather forecasts often use inequalities to describe temperature ranges. For instance, a forecast might say the temperature will be between 70°F and 80°F. This can be written as $70 ≤ temperature ≤ 80$.

5. Capacity Limits

Think about the maximum weight a bridge can hold or the maximum number of people allowed in an elevator. These are capacity limits that can be expressed as inequalities.

Practice Problems

Alright, now it's your turn to shine! Let's try a few practice problems to solidify your understanding of graphing inequalities on the number line.

Practice Problem 1

Graph the solution to the inequality $4x + 2 < 10$ on the number line.

Practice Problem 2

Graph the solution to the inequality $-3(x - 1) ≥ 6$ on the number line.

Practice Problem 3

Graph the solution to the inequality $5 - 2x > 1$ on the number line.

Take your time, work through each step carefully, and remember to check your answers. The more you practice, the more confident you'll become in solving and graphing inequalities!

Conclusion

And there you have it! We've walked through how to solve the inequality $6 - 2(x - 3) > 3x - 3$ and graph its solution on a number line. Remember, the key is to simplify the inequality step by step, isolate the variable, and then represent the solution visually on the number line.

Graphing inequalities might seem a bit tricky at first, but with practice, you'll become a pro in no time. Inequalities are a fundamental concept in math, and understanding them will open up doors to more advanced topics. Plus, as we've seen, they have tons of real-world applications, from budgeting to understanding speed limits.

So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!