What Is The Solution To The Inequality $5 - 3(2x + 5) \leq 8$?A. $k \geq \frac{1}{3}$ B. $k \leq \frac{1}{3}$ C. $k \geq -3$ D. $k \leq -3$

Introduction

In this article, we will explore the solution to the given inequality $5 - 3(2x + 5) \leq 8$. To solve this inequality, we will use the distributive property, combine like terms, and isolate the variable. We will also provide a step-by-step solution to help readers understand the process.

Step 1: Apply the Distributive Property

The first step in solving the inequality is to apply the distributive property to the expression $-3(2x + 5)$. This will allow us to simplify the expression and make it easier to work with.

$$5 - 3(2x + 5) \leq 8$$

Using the distributive property, we can rewrite the expression as:

$$5 - 6x - 15 \leq 8$$

Step 2: Combine Like Terms

Next, we will combine like terms to simplify the expression further. We can combine the constants $5$ and $-15$ to get:

$$-6x - 10 \leq 8$$

Step 3: Isolate the Variable

Now, we will isolate the variable $x$ by adding $10$ to both sides of the inequality. This will give us:

$$-6x \leq 18$$

Step 4: Solve for x

To solve for $x$, we will divide both sides of the inequality by $-6$. However, since we are dividing by a negative number, we must reverse the direction of the inequality. This will give us:

$$x \geq -3$$

Conclusion

In conclusion, the solution to the inequality $5 - 3(2x + 5) \leq 8$ is $x \geq -3$. This means that any value of $x$ that is greater than or equal to $-3$ will satisfy the inequality.

Discussion

The solution to the inequality $5 - 3(2x + 5) \leq 8$ is a simple one. However, it requires careful application of the distributive property, combining like terms, and isolating the variable. By following these steps, we can solve even the most complex inequalities.

Common Mistakes

When solving inequalities, it's easy to make mistakes. One common mistake is to forget to reverse the direction of the inequality when dividing by a negative number. Another mistake is to forget to combine like terms. By being careful and following the steps, we can avoid these mistakes and find the correct solution.

Real-World Applications

Inequalities are used in many real-world applications, such as finance, engineering, and science. For example, in finance, inequalities are used to model the growth of investments and the risk of default. In engineering, inequalities are used to design and optimize systems. In science, inequalities are used to model the behavior of complex systems.

Final Answer

The final answer to the inequality $5 - 3(2x + 5) \leq 8$ is $x \geq -3$. This means that any value of $x$ that is greater than or equal to $-3$ will satisfy the inequality.

Frequently Asked Questions

• Q: What is the solution to the inequality $5 - 3(2x + 5) \leq 8$? A: The solution to the inequality is $x \geq -3$.
• Q: How do I solve an inequality? A: To solve an inequality, you must apply the distributive property, combine like terms, and isolate the variable.
• Q: What are some common mistakes when solving inequalities? A: Some common mistakes include forgetting to reverse the direction of the inequality when dividing by a negative number and forgetting to combine like terms.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Inequalities" by Michael Artin

Conclusion

In conclusion, the solution to the inequality $5 - 3(2x + 5) \leq 8$ is $x \geq -3$. This means that any value of $x$ that is greater than or equal to $-3$ will satisfy the inequality. By following the steps outlined in this article, readers can solve even the most complex inequalities.