Solving The Inequality: 2 - 6(8 - M) ≤ 8m - 38

Hey guys! Let's dive into solving this inequality step-by-step. Inequalities might seem intimidating, but with a bit of algebraic manipulation, they become pretty manageable. So, grab your pencils, and let's get started!

Understanding the Inequality

Our mission, should we choose to accept it, is to solve the inequality: $2 - 6(8 - m) leq 8m - 38$. Solving an inequality involves finding the range of values for the variable (in this case, 'm') that makes the inequality true. It's similar to solving an equation, but instead of finding a single value, we're often finding a set of values.

Step 1: Distribute the -6

The first thing we need to do is simplify the expression by distributing the -6 across the terms inside the parenthesis. This means we multiply -6 by both 8 and -m:

$2 - 6(8) - 6(-m) leq 8m - 38$

Which simplifies to:

$2 - 48 + 6m leq 8m - 38$

So far so good, right? We're just cleaning things up so we can see the problem more clearly. This distribution step is crucial because it removes the parentheses, allowing us to combine like terms more easily. Remember, always pay attention to the signs when distributing negative numbers!

Step 2: Combine Like Terms

Now, let's combine the constant terms on the left side of the inequality:

$2 - 48 = -46$

So our inequality now looks like this:

$-46 + 6m leq 8m - 38$

Combining like terms helps to consolidate the constants, which simplifies the equation and makes it easier to isolate the variable. Keep an eye out for any other like terms you can combine in more complex problems. This step is all about making things as simple as possible.

Step 3: Isolate the Variable Term

Next, we want to get all the 'm' terms on one side of the inequality. Let's subtract 6m from both sides to move the 'm' term from the left to the right:

$-46 + 6m - 6m leq 8m - 38 - 6m$

Which simplifies to:

$-46 leq 2m - 38$

Moving variable terms to one side is a fundamental step in solving inequalities. By doing this, we can then isolate the variable and find its possible values. Make sure you perform the same operation on both sides to maintain the balance of the inequality.

Step 4: Isolate the Variable

Now, let's isolate 'm' by adding 38 to both sides of the inequality:

$-46 + 38 leq 2m - 38 + 38$

Which simplifies to:

$-8 leq 2m$

Step 5: Solve for m

Finally, to solve for 'm', we need to divide both sides of the inequality by 2:

$\frac{-8}{2} leq \frac{2m}{2}$

Which gives us:

$-4 leq m$

So, we find that $m ge -4$. This means that any value of 'm' greater than or equal to -4 will satisfy the original inequality. We've done it!

Writing the Solution

The solution to the inequality $2 - 6(8 - m) leq 8m - 38$ is $m ge -4$. In interval notation, this is written as $[-4, \infty)$. This means that 'm' can be any number from -4 (inclusive) to positive infinity. Remember, the square bracket indicates that -4 is included in the solution, while the parenthesis indicates that infinity is not a specific number and cannot be included.

Verification

To ensure our solution is correct, let's test a value of 'm' that is greater than or equal to -4. How about $m = 0$?

Plugging $m = 0$ into the original inequality gives us:

$2 - 6(8 - 0) leq 8(0) - 38$

$2 - 6(8) leq -38$

$2 - 48 leq -38$

$-46 leq -38$

This is true, so our solution is likely correct. Now let's test $m = -4$:

$2 - 6(8 - (-4)) leq 8(-4) - 38$

$2 - 6(12) leq -32 - 38$

$2 - 72 leq -70$

$-70 leq -70$

This is also true. So, it looks like we nailed it!

Common Mistakes to Avoid

1. Forgetting to Distribute: Always make sure to distribute the number outside the parentheses to all terms inside. This is a classic mistake that can throw off your entire solution.
2. Incorrectly Combining Like Terms: Double-check your addition and subtraction when combining like terms. A small arithmetic error can lead to a wrong answer.
3. Not Flipping the Inequality Sign: If you multiply or divide both sides of an inequality by a negative number, you need to flip the inequality sign. This is a crucial rule to remember. Since we only added, subtracted, and divided by a positive number we didn't have to worry about this, but always keep it in mind.
4. Misinterpreting the Solution: Make sure you understand what your solution means. For example, if you get $m > 3$, it means 'm' can be any number greater than 3, but not 3 itself.

Why Inequalities Matter

Inequalities are used everywhere in the real world! They pop up in science, engineering, economics, and even in everyday decision-making. For example, you might use an inequality to determine how many hours you need to work to earn enough money to buy something you want, or to figure out the minimum speed you need to drive to arrive at your destination on time. Understanding inequalities helps you make informed decisions and solve real-world problems. Inequalities can be used to define constraints. For example: A company might have a budget constraint, like its spending must be less than or equal to X dollars. Or you might have constraints based on time. Inequalities are used in the definition of domain of the function. When dealing with square roots, you have to make sure that the expression under the square root is non-negative.

Practice Problems

Want to sharpen your skills? Try solving these inequalities:

1. $3x + 5 > 14$
2. $-2(y - 1) leq 6$
3. $4a - 7 < 9 + 2a$

Solving inequalities is a fundamental skill in algebra. By understanding the steps involved and practicing regularly, you'll become more confident and proficient in solving them. So keep practicing, and don't be afraid to ask for help when you need it.

Conclusion

Alright, guys, we've reached the end of our inequality-solving journey! We tackled the problem step-by-step, from distributing and combining like terms to isolating the variable and interpreting the solution. Remember, practice makes perfect, so keep honing your skills! Until next time, keep those pencils sharp and those minds even sharper. You've got this! See you in the next mathematical adventure! Keep an eye out for more breakdowns, explainers, and fun with numbers.