Solving Inequality: Step-by-Step Guide For Z

Hey guys! Today, we're diving into the world of inequalities and tackling a common problem: solving for a variable. Specifically, we're going to break down how to solve the inequality $10-2(8z-5) \_geq -10z-8-10z$ for $z$ and express the solution in its simplest form. Inequalities might seem a bit intimidating at first, but trust me, with a step-by-step approach, you'll be solving them like a pro in no time. So, let's get started!

Understanding Inequalities

Before we jump into the problem, let's quickly recap what inequalities are. Unlike equations that have a single solution, inequalities represent a range of possible solutions. Think of it like this: instead of saying $x$ equals 5, we're saying $x$ is greater than, less than, greater than or equal to, or less than or equal to a certain value. These relationships are expressed using symbols like > (greater than), < (less than), $\_geq$ (greater than or equal to), and $\_leq$ (less than or equal to).

The key thing to remember is that when we perform operations on inequalities, we need to be mindful of how those operations affect the inequality sign. For example, multiplying or dividing both sides by a negative number flips the direction of the inequality. We'll see this in action as we solve our problem.

Why Inequalities Matter

You might be wondering, "Why should I care about inequalities?" Well, inequalities are super useful in real-world situations where we're dealing with ranges or constraints. For example, you might use inequalities to determine the minimum score you need on a test to get a certain grade, or to figure out how many items you can buy within a specific budget. They're also fundamental in various fields like economics, engineering, and computer science.

Step-by-Step Solution

Okay, let's get down to business and solve the inequality $10-2(8z-5) \_geq -10z-8-10z$. We'll break it down into manageable steps so you can follow along easily.

Step 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 parentheses:

$10 - 2(8z - 5) \_geq -10z - 8 - 10z$

$10 - 16z + 10 \_geq -10z - 8 - 10z$

Notice that multiplying -2 by -5 gives us +10. It's crucial to pay attention to the signs to avoid making mistakes.

Step 2: Combine Like Terms

Now, let's simplify both sides of the inequality by combining like terms. On the left side, we have 10 and 10, which add up to 20. On the right side, we have -10z and -10z, which combine to -20z:

$20 - 16z \_geq -20z - 8$

Step 3: Isolate the Variable Term

Our goal is to get all the $z$ terms on one side of the inequality and the constants on the other. Let's add 20z to both sides to eliminate the -20z on the right side:

$20 - 16z + 20z \_geq -20z - 8 + 20z$

$20 + 4z \_geq -8$

Step 4: Isolate the Constant Term

Next, we want to isolate the $z$ term by getting rid of the 20 on the left side. We can do this by subtracting 20 from both sides:

$20 + 4z - 20 \_geq -8 - 20$

$4z \_geq -28$

Step 5: Solve for z

Finally, to solve for $z$, we need to divide both sides by 4:

rac{4z}{4} \_geq rac{-28}{4}

$z \_geq -7$

And there you have it! The solution to the inequality is $z \_geq -7$. This means that any value of $z$ that is greater than or equal to -7 will satisfy the original inequality.

Expressing the Solution

We've solved for $z$, but let's talk about how to express the solution in different ways. This is important because you might be asked to provide the answer in a specific format.

Inequality Notation

We've already expressed the solution in inequality notation: $z \_geq -7$. This is a clear and concise way to represent the solution.

Number Line

Another way to represent the solution is on a number line. To do this, we draw a number line and mark -7 on it. Since $z$ is greater than or equal to -7, we'll use a closed circle (or a square bracket) at -7 to indicate that -7 is included in the solution. Then, we draw an arrow extending to the right, indicating all values greater than -7.

Interval Notation

Interval notation is a compact way to represent a range of values. For the solution $z \_geq -7$, we use a square bracket to indicate that -7 is included and a parenthesis for infinity. The interval notation for this solution is $[-7, \infty)$.

• A square bracket [ or ] indicates that the endpoint is included in the interval.
• A parenthesis ( or ) indicates that the endpoint is not included in the interval.
• Infinity ($\infty$) and negative infinity ($-\infty$) always use parentheses because they are not actual numbers and cannot be included in the interval.

Common Mistakes to Avoid

When solving inequalities, there are a few common mistakes that students often make. Let's go over these so you can avoid them.

Forgetting to Flip the Inequality Sign

The most common mistake is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. Remember, if you multiply or divide by a negative, you need to reverse the direction of the inequality.

Incorrectly Distributing

Another common mistake is making errors during the distribution step. Pay close attention to the signs and make sure you multiply the term outside the parentheses by every term inside the parentheses.

Combining Unlike Terms

Be careful not to combine terms that are not like terms. For example, you can't combine a constant with a term containing a variable.

Misinterpreting the Solution

Make sure you understand what the solution means. For example, $z \_geq -7$ means that $z$ can be -7 or any number greater than -7.

Practice Problems

To really master solving inequalities, you need to practice! Here are a few practice problems for you to try:

1. $3(x + 2) < 5x - 4$
2. $-2(2y - 1) \_leq 10$
3. $4 - (3z + 1) > 2z + 7$

Work through these problems step-by-step, and don't forget to check your answers. The more you practice, the more confident you'll become.

Conclusion

Solving inequalities might seem tricky at first, but with a clear understanding of the steps and some practice, you can conquer any inequality that comes your way. Remember to distribute, combine like terms, isolate the variable, and be mindful of flipping the inequality sign when multiplying or dividing by a negative number.

By following these steps and avoiding common mistakes, you'll be able to solve inequalities with confidence. And remember, if you ever get stuck, don't hesitate to review the steps or ask for help. Keep practicing, and you'll become an inequality-solving master in no time! Keep up the great work, guys!