Solve For $z$.$3z \ \textgreater \ 6$

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Solving Inequalities: A Step-by-Step Guide to Solving for z in the Inequality 3z > 6

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving the inequality 3z > 6, where z is the variable we need to solve for.

Understanding the Inequality

The given inequality is 3z > 6. This means that the product of 3 and z is greater than 6. To solve for z, we need to isolate the variable z on one side of the inequality.

Step 1: Divide Both Sides by 3

To isolate z, we need to get rid of the coefficient 3 that is being multiplied by z. We can do this by dividing both sides of the inequality by 3.

\frac{3z}{3} > \frac{6}{3}

Simplifying the Inequality

When we divide both sides of the inequality by 3, we get:

z > 2

Interpreting the Solution

The solution to the inequality 3z > 6 is z > 2. This means that z can take any value greater than 2. In other words, z can be 2.1, 2.5, 3, 3.5, and so on.

Visualizing the Solution

To visualize the solution, we can plot the inequality on a number line. The number line represents all possible values of z. The inequality z > 2 indicates that z can take any value to the right of 2 on the number line.

In conclusion, solving the inequality 3z > 6 involves isolating the variable z on one side of the inequality. By dividing both sides of the inequality by 3, we get z > 2. This means that z can take any value greater than 2. The solution to the inequality can be visualized on a number line, where z can take any value to the right of 2.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid common mistakes. Here are a few:

  • Not isolating the variable: Make sure to isolate the variable on one side of the inequality.
  • Not considering the direction of the inequality: Pay attention to the direction of the inequality (greater than, less than, or equal to).
  • Not checking the solution: Always check the solution to ensure it satisfies the original inequality.

Real-World Applications

Solving inequalities has numerous real-world applications. Here are a few examples:

  • Finance: In finance, inequalities are used to calculate interest rates, investment returns, and loan payments.
  • Science: In science, inequalities are used to model population growth, chemical reactions, and physical systems.
  • Engineering: In engineering, inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Solving inequalities is a fundamental concept in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can solve inequalities with confidence. Remember to isolate the variable, consider the direction of the inequality, and check the solution. With practice and patience, you'll become proficient in solving inequalities and apply them to real-world problems.