Solving Inequalities: A Step-by-Step Guide

Let's dive into solving the inequality $2(4+2x) \geq 5x+5$. Inequalities are a fundamental concept in mathematics, and mastering them is super important for various applications. In this guide, we will break down each step to ensure you understand the process thoroughly. Understanding the step-by-step solutions and the logic behind them is more important than just memorizing formulas, so let's get started!

Understanding the Inequality

Before we start crunching numbers, let's break down what this inequality actually means. The expression $2(4+2x) \geq 5x+5$ is telling us that whatever value we get when we compute $2(4+2x)$, it must be greater than or equal to the value we get when we compute $5x+5$. Our mission is to find all the values of x that make this statement true. Inequalities are used everywhere, from optimizing business costs to understanding physical constraints in engineering. So, understanding inequalities isn't just about solving math problems; it's about applying math to the real world.

The greater than or equal to sign, "$\geq$", is crucial. It means that the left side can be equal to the right side or larger than it. This contrasts with a strict inequality like "$>$", which means strictly greater than.

Step-by-Step Solution

1. Distribute

The first thing we need to do is simplify both sides of the inequality. Let's start by distributing the 2 on the left side of the inequality: $2(4+2x)$.

$2 * 4 = 8$ $2 * 2x = 4x$

So, $2(4+2x)$ becomes $8 + 4x$. Our inequality now looks like this: $8 + 4x \geq 5x + 5$.

Distribution helps to remove parentheses, which makes the equation easier to handle. Remember, the goal is to isolate x on one side of the inequality.

2. Combine Like Terms

Next, we want to gather all the x terms on one side and the constants on the other. A good way to do this is to subtract $4x$ from both sides. This keeps the inequality balanced.

$8 + 4x - 4x \geq 5x + 5 - 4x$

This simplifies to:

$8 \geq x + 5$

Now, let's subtract 5 from both sides to isolate the x term further:

$8 - 5 \geq x + 5 - 5$

Which gives us:

$3 \geq x$

3. Isolate the Variable

We've now arrived at $3 \geq x$. This is the same as saying $x \leq 3$. Both notations mean that x can be any value less than or equal to 3. This is our solution!

So, to recap, we distributed, combined like terms, and isolated the variable. These are standard techniques for solving inequalities, and with practice, you'll become super quick at them.

4. Understanding the Solution

The solution $x \leq 3$ means that any number that is 3 or less will satisfy the original inequality. For example, if we plug in $x = 0$:

$2(4 + 2(0)) \geq 5(0) + 5$ $2(4) \geq 5$ $8 \geq 5$ (True)

And if we plug in $x = 3$:

$2(4 + 2(3)) \geq 5(3) + 5$ $2(4 + 6) \geq 15 + 5$ $2(10) \geq 20$ $20 \geq 20$ (True)

If we were to pick a number greater than 3, like $x = 4$:

$2(4 + 2(4)) \geq 5(4) + 5$ $2(4 + 8) \geq 20 + 5$ $2(12) \geq 25$ $24 \geq 25$ (False)

This shows that our solution is indeed correct.

Choosing the Correct Answer

Looking back at the options provided:

A. $x \leq -2$ B. $x \geq -2$ C. $x \leq 3$ D. $x \geq 3$

The correct answer is C. $x \leq 3$.

Tips for Solving Inequalities

Here are a few extra tips to keep in mind when solving inequalities:

• Always distribute carefully: Make sure you multiply each term inside the parentheses correctly.
• Combine like terms accurately: Pay attention to the signs (+ and -) when combining terms.
• When multiplying or dividing by a negative number, flip the inequality sign: For example, if you have $-x > 2$, multiplying by -1 gives you $x < -2$.
• Check your answer: Plug your solution back into the original inequality to make sure it holds true.
• Practice, practice, practice: The more you practice, the better you'll get at solving inequalities.

Common Mistakes to Avoid

• Forgetting to flip the inequality sign: As mentioned above, this is a common mistake when multiplying or dividing by a negative number.
• Incorrect distribution: Double-check that you've multiplied each term inside the parentheses correctly.
• Arithmetic errors: Small mistakes in arithmetic can lead to incorrect solutions.
• Not checking the answer: Always plug your solution back into the original inequality to verify its correctness.

Real-World Applications

Inequalities aren't just abstract math concepts; they have tons of real-world applications. For example:

• Budgeting: You might use inequalities to determine how much you can spend on different items while staying within your budget.
• Engineering: Engineers use inequalities to design structures that can withstand certain loads or to optimize the performance of a system.
• Business: Businesses use inequalities to maximize profits or minimize costs.
• Health: Doctors might use inequalities to determine the appropriate dosage of a medication.

Practice Problems

To solidify your understanding, try solving these practice problems:

1. $3(2x - 1) < 4x + 5$
2. $-2(x + 3) \geq 6 - x$
3. $5x - 7 \leq 2x + 2$

Work through these problems step-by-step, and remember to check your answers. Good luck!

Conclusion

So there you have it! We've successfully solved the inequality $2(4+2x) \geq 5x+5$. Remember the key steps: distribute, combine like terms, and isolate the variable. Inequalities are a powerful tool in mathematics and have widespread applications in various fields. By following the steps and tips outlined in this guide, you can confidently tackle any inequality that comes your way. Keep practicing, and you'll become a pro in no time! Whether you are dealing with basic algebra or advanced calculus, mastering inequalities is a must. Keep up the great work!