Problem 41 - Prealgebra

Question

Solve each inequality. Check your solution. $$z-2>-3.8$$

Step-by-Step Solution

Answer

The solution is \(z > -1.8\).

Step 1: Isolate the variable

To isolate the variable, we need to get \(z\) by itself on one side of the inequality. The inequality given is \(z - 2 > -3.8\). To do this, we need to add 2 to both sides: \[z - 2 + 2 > -3.8 + 2\]This simplifies to:\[z > -1.8\]

Step 2: Check the solution

To verify the solution, choose a number greater than \(-1.8\) and see if it satisfies the original inequality. Let's try \(z = 0\): Substitute \(z = 0\) into the inequality: \[0 - 2 > -3.8\]Which simplifies to: \[-2 > -3.8\]Since this statement is true, the solution \(z > -1.8\) is correct.

Key Concepts

Problem Solving

When tackling any mathematical exercise, including inequalities, problem-solving skills are essential. The first step in solving an inequality involves understanding what the inequality is asking for. In the exercise, you are asked to solve the inequality \( z - 2 > -3.8 \). This means finding all values of \( z \) that make the inequality true.

To do this, we begin by isolating the variable. In the given inequality, \( z \) is the variable, and our goal is to get \( z \) alone on one side. This involves performing operations that simplify the inequality while also maintaining its balance.

It’s crucial to remember that when you perform an operation on one side of an inequality, you must do the same to the other side. This ensures the inequality's integrity. For example, adding or subtracting the same value from both sides is a basic method used to isolate the variable, as seen in our example: add 2 to both sides of the inequality which gives us \( z > -1.8 \).

By practicing these types of exercises, you enhance your problem-solving skills, which are critical tools not just in math, but in real-life scenarios as well. Problem-solving involves logical thinking and applying systematic methods to find solutions.

Checking Solutions

Once you've found a potential solution to your inequality, the next step is checking to see if it truly satisfies the inequality. This verification process is crucial for ensuring the correctness of your solution.

In our example, after solving for \( z \) and finding \( z > -1.8 \), a simple way to verify this solution is by picking any value greater than \(-1.8\) and substituting it back into the original inequality to check if it holds true.

For instance:

• Choose a number greater than \(-1.8\), such as 0.
• Substitute \( z = 0 \) into the inequality: \( 0 - 2 > -3.8 \).
• This simplifies to \( -2 > -3.8 \), which is a true statement.

This shows that \( z = 0 \) satisfies the original inequality, confirming that our solution \( z > -1.8 \) is indeed correct. Hence, checking solutions not only supports the correctness of your answer but also builds confidence in your understanding of the steps taken during problem solving.

Step-by-Step Solutions

Breaking down problems into step-by-step solutions is a powerful method to simplify and tackle inequalities effectively. This approach helps in visualizing each part of the problem separately, making it easier to follow and understand.

For the inequality \( z - 2 > -3.8 \):

• Step 1: Isolate the variable. To solve the inequality, we start by simplifying. Add 2 to both sides, resulting in the inequality \( z > -1.8 \).
• Step 2: Verify the solution. Choose a value, like 0, that is greater than \(-1.8\) and check it against the original inequality. It should hold true to confirm that your solution is correct.

Step-by-step solutions are particularly beneficial because they prevent errors and alert you to check every operation's effect on your inequality. Additionally, it aids in clarity and ensures you don't rush through important calculations that could affect the final outcome. Following steps meticulously can significantly improve your ability to solve more complex problems with confidence.