Solving & Graphing: 6x - 6 ≥ 7x - 2x Inequality

Hey everyone! Today, let's tackle a fun little math problem: solving the inequality $6x - 6 \geq 7x - 2x$ and graphing its solution set. Inequalities might seem intimidating at first, but with a few simple steps, we can break them down and find the values of 'x' that make the statement true. So, grab your pencils, and let's dive in!

Step 1: Simplify the Inequality

Our first goal is to simplify both sides of the inequality as much as possible. This makes the problem easier to manage. We start with:

$6x - 6 \geq 7x - 2x$

Notice that on the right side, we have $7x - 2x$. These are like terms, meaning they both contain the variable 'x' raised to the same power (in this case, the power is 1). We can combine these like terms:

$7x - 2x = 5x$

So, our inequality now looks like this:

$6x - 6 \geq 5x$

Great! We've simplified the right side. Now both sides are as simple as they can be. This sets us up nicely for the next step, where we isolate the variable 'x'. Combining like terms is a fundamental skill in algebra, and mastering it will help you solve all sorts of equations and inequalities more efficiently. Remember, you can only combine terms that have the exact same variable part. For example, you can combine $3x$ and $5x$ to get $8x$, but you can't combine $3x$ and $5x^2$ because the powers of 'x' are different. Keep an eye out for like terms, and always simplify before moving on to more complicated steps.

Step 2: Isolate the Variable

Now we want to get all the 'x' terms on one side of the inequality and all the constant terms (the numbers without 'x') on the other side. Let's subtract $5x$ from both sides of the inequality:

$6x - 6 - 5x \geq 5x - 5x$

This simplifies to:

$x - 6 \geq 0$

Next, we want to isolate 'x' completely. To do this, we add 6 to both sides of the inequality:

$x - 6 + 6 \geq 0 + 6$

This gives us:

$x \geq 6$

Fantastic! We've successfully isolated 'x'. Our solution tells us that 'x' must be greater than or equal to 6 for the original inequality to be true. Isolating the variable is a crucial step in solving any equation or inequality. The key is to perform the same operation on both sides to maintain the balance. Think of it like a scale – whatever you add or subtract from one side, you must also add or subtract from the other to keep it level. This ensures that the relationship between the two sides remains unchanged. Whether you're dealing with simple linear equations or more complex inequalities, mastering the art of isolating variables is essential for finding the solution.

Step 3: Graph the Solution Set

Now that we have our solution, $x \geq 6$, we need to graph it on a number line. This will give us a visual representation of all the possible values of 'x' that satisfy the inequality.

1. Draw a Number Line: Start by drawing a straight line and marking some numbers on it. Make sure to include 6, as it's the boundary of our solution.

2. Mark the Boundary Point: Since our inequality is $x \geq 6$ (greater than or equal to), we use a closed circle (or a filled-in dot) at 6. This indicates that 6 is included in the solution set. If the inequality were $x > 6$ (strictly greater than), we would use an open circle to show that 6 is not included.

3. Shade the Correct Region: We want to represent all values of 'x' that are greater than or equal to 6. On the number line, these values are to the right of 6. So, we shade the region to the right of the closed circle at 6, extending the shading indefinitely in that direction. This shaded region represents all the numbers that satisfy the inequality $x \geq 6$.

By graphing the solution set, we gain a clear visual understanding of the inequality's solution. The closed circle at 6 and the shaded region to the right tell us that any number 6 or larger will make the original inequality true. Graphing solution sets is a valuable tool for visualizing mathematical relationships and can be particularly helpful when dealing with more complex inequalities or systems of inequalities.

Step 4: Verification (Optional but Recommended)

To be absolutely sure our solution is correct, we can test a value within our solution set and a value outside of it in the original inequality.

• Test a value within the solution set: Let's pick $x = 7$ (since $7 \geq 6$). Plug it into the original inequality:

  $6(7) - 6 \geq 7(7) - 2(7)$

  $42 - 6 \geq 49 - 14$

  $36 \geq 35$ This is true, so our solution is likely correct.

• Test a value outside the solution set: Let's pick $x = 5$ (since $5 < 6$). Plug it into the original inequality:

  $6(5) - 6 \geq 7(5) - 2(5)$

  $30 - 6 \geq 35 - 10$

  $24 \geq 25$ This is false, which further confirms our solution.

Verifying your solution is like double-checking your work. It helps catch any mistakes you might have made along the way and gives you confidence that your answer is correct. By plugging in values both inside and outside the solution set, you can ensure that the inequality holds true for the values within the set and false for the values outside of it. This step is especially useful when dealing with more complex inequalities where errors are more likely to occur.

Conclusion

So, to wrap it up, we solved the inequality $6x - 6 \geq 7x - 2x$ and found that $x \geq 6$. We then graphed this solution on a number line, showing a closed circle at 6 and shading to the right. Remember, the key to solving inequalities is to simplify, isolate the variable, and then visualize the solution set. With practice, you'll become a pro at solving inequalities! Keep up the great work, and happy problem-solving!