Solve For $z$.$10z \geq -10$

Feb 25, 2025 by ADMIN 29 views

Solving Inequalities: A Step-by-Step Guide to Finding the Value of $z$ in the Inequality $10z \geq -10$

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems in algebra, geometry, and other branches of mathematics. An inequality is a statement that two expressions are not equal, but one is greater than or less than the other. In this article, we will focus on solving the inequality $10z \geq -10$ to find the value of $z$ . We will break down the solution into simple steps, making it easy to understand and follow.

The given inequality is $10z \geq -10$ . This means that the product of $10$ and $z$ is greater than or equal to $-10$ . To solve this inequality, we need to isolate the variable $z$ .

Step 1: Divide Both Sides by 10

To isolate $z$ , we need to get rid of the coefficient $10$ that is being multiplied by $z$ . We can do this by dividing both sides of the inequality by $10$ . This will give us:

z \geq \frac{-10}{10}

Step 2: Simplify the Right-Hand Side

Now, we need to simplify the right-hand side of the inequality. We can do this by dividing $-10$ by $10$ , which gives us:

z \geq -1

Step 3: Write the Solution in Interval Notation

The solution to the inequality $10z \geq -10$ is $z \geq -1$ . We can write this in interval notation as $[-1, \infty)$ .

In this article, we solved the inequality $10z \geq -10$ to find the value of $z$ . We broke down the solution into simple steps, making it easy to understand and follow. We started by dividing both sides of the inequality by $10$ to isolate $z$ , and then simplified the right-hand side to get the final solution. The solution to the inequality is $z \geq -1$ , which can be written in interval notation as $[-1, \infty)$ .

Why is Solving Inequalities Important?

Solving inequalities is an essential skill in mathematics that has numerous applications in real-life situations. Inequalities are used to model real-world problems, such as:

Finance: Inequalities are used to calculate interest rates, investment returns, and other financial metrics.
Science: Inequalities are used to model population growth, chemical reactions, and other scientific phenomena.
Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Tips and Tricks for Solving Inequalities

Here are some tips and tricks for solving inequalities:

Read the inequality carefully: Make sure you understand what the inequality is saying.
Use inverse operations: Use inverse operations, such as addition and subtraction, to isolate the variable.
Check your solution: Check your solution by plugging it back into the original inequality.
Use interval notation: Use interval notation to write the solution in a concise and clear way.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving inequalities:

Not reading the inequality carefully: Make sure you understand what the inequality is saying.
Not using inverse operations: Use inverse operations, such as addition and subtraction, to isolate the variable.
Not checking your solution: Check your solution by plugging it back into the original inequality.
Not using interval notation: Use interval notation to write the solution in a concise and clear way.

In our previous article, we solved the inequality $10z \geq -10$ to find the value of $z$ . We broke down the solution into simple steps, making it easy to understand and follow. In this article, we will provide a Q&A guide to help you better understand solving inequalities.

Q: What is an inequality?

A: An inequality is a statement that two expressions are not equal, but one is greater than or less than the other.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by using inverse operations, such as addition and subtraction.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality that can be written in the form $ax + b \geq c$ , where $a$ , $b$ , and $c$ are constants. A quadratic inequality is an inequality that can be written in the form $ax^2 + bx + c \geq 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable by using inverse operations, such as addition and subtraction.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict inequality symbol, such as {{content}}lt; $ or {{content}}gt; $. A non-strict inequality is an inequality that is written with a non-strict inequality symbol, such as $\leq$ or $\geq$ .

Q: How do I determine the solution to an inequality?

A: To determine the solution to an inequality, you need to isolate the variable and then use the sign of the inequality to determine the solution.

Q: What is the importance of solving inequalities?

A: Solving inequalities is an essential skill in mathematics that has numerous applications in real-life situations. Inequalities are used to model real-world problems, such as finance, science, and engineering.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

Not reading the inequality carefully
Not using inverse operations
Not checking your solution
Not using interval notation

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through examples and exercises in your textbook or online resources. You can also try solving inequalities on your own and then check your solution with a calculator or online tool.

Solving inequalities is an essential skill in mathematics that has numerous applications in real-life situations. In this article, we provided a Q&A guide to help you better understand solving inequalities. We discussed the importance of solving inequalities, provided tips and tricks for solving inequalities, and highlighted common mistakes to avoid. We also provided examples and exercises to help you practice solving inequalities.

If you need additional help with solving inequalities, here are some additional resources that you can use:

Textbooks: You can find textbooks on algebra and mathematics that cover solving inequalities.
Online resources: You can find online resources, such as Khan Academy and Mathway, that provide video lessons and interactive exercises on solving inequalities.
Tutors: You can find tutors who can provide one-on-one help with solving inequalities.
Practice problems: You can find practice problems on solving inequalities that you can work through on your own.