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When we solve an equation, the solution might be a negative number. A negative solution simply means that the value of the variable is less than zero. In the given exercise, we use a negative solution of -150.

This concept is crucial in understanding that solutions are not limited to positive numbers or zero. Negative solutions can arise frequently in real-life scenarios, such as when dealing with debt or below-ground levels.

To create an equation with a negative solution, start by thinking of a number less than zero as the variable's value. In this example, we chose -150 as the solution of the equation we aimed to create. Ensuring that students understand the occurrence of negative solutions will allow them to solve a variety of problems more confidently.

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. This is a powerful tool for solving all kinds of algebraic problems.

In the problem provided, we started with the equation \( x = -150 \). This equation clearly presents -150 as the solution for \( x \). However, we are tasked with manipulating it so that adding 100 to both sides doesn't affect equality, eventually leading to the same solution.

The simplicity of algebraic manipulation lies in maintaining equality while rearranging components, such as constants and variables. For example, when we added 100 to both sides of the equation \( x + 100 = -150 + 100 \), we maintained the balance of the equation while changing its form.

• Keep the balance: Any operation you perform on one side must also be performed on the other side.
• Rearrange the terms: Organize terms to isolate variables, making them stand out for easy solving.
• Simplify: Reduce equations to the simplest form, turning complex expressions into manageable ones.

This exercise showcases a fundamental algebraic tactic to reach simpler and logically equivalent equations.

Inverse operations are crucial for solving equations as they allow us to 'undo' mathematical actions. For example, if an operation adds a number, its inverse operation would be subtraction, and vice versa.

In solving equations, using inverse operations helps isolate the variable we're solving for. For instance, let's look at \( x + 100 = -50 \). To find the value of \( x \), you'll perform the inverse of adding 100, which is subtracting 100 from both sides of the equation. This results in:

\[ x + 100 - 100 = -50 - 100 \]

Simplifying further gives:

\[ x = -150 \]

Through inverse operations, we reached our original solution for \( x \).

• Addition and subtraction are inverse operations.
• Multiplication and division are another pair of inverse operations.
• This concept helps in systematically solving and simplifying equations.

By understanding and applying inverse operations, students can confidently solve equations and verify their solutions.