Problem 8 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$-\frac{1}{2} x=-4$$

Step-by-Step Solution

Answer

The solution is \(x = 8\).

Step 1: Understand the Problem

The equation given is \(-\frac{1}{2} x = -4\). Our task is to solve for \(x\) using the multiplication property of equality. This property states that if you multiply both sides of an equation by the same nonzero number, the two sides remain equal.

Step 2: Isolate x using Multiplication

To isolate \(x\), we need to eliminate the fraction by multiplying both sides of the equation by the reciprocal of \(-\frac{1}{2}\), which is \(-2\). This step ensures that the coefficient of \(x\) becomes 1, making it easy to solve for \(x\).

Step 3: Multiply Both Sides

Multiply both sides by \(-2\):\[(-2) \times \left( -\frac{1}{2} x \right) = (-2) \times (-4)\]This simplifies to:\[x = 8\]

Step 4: Verify the Solution (Optional)

To ensure the solution is correct, substitute \(x = 8\) back into the original equation:\[-\frac{1}{2}(8) = -4\]Simplifying the left side gives \(-4 = -4\), confirming that our solution is correct.

Key Concepts

Solving Equations

Equation solving is a fundamental aspect of algebra that helps us find unknown values. When we solve equations, we essentially balance two sides of an equation by performing operations such as addition, subtraction, multiplication, or division on both sides. This ensures the equation stays true. In our problem

• We start with an equation: \(-\frac{1}{2} x = -4\).
• The goal is to determine what value of \(x\) makes this equation true.
• We use the multiplication property of equality, which helps keep the balance while isolating the variable.

Understanding the principles of solving equations is crucial for moving up to more advanced math, as well as applying math in real-life situations.

Prealgebra

Prealgebra is an important stepping stone between basic arithmetic and more advanced algebra. It involves working with simple equations and inequalities, like the one in this exercise:

• We learn to manipulate integers, fractions, and decimals comfortably.
• There is a focus on understanding properties like the multiplication property of equality, used here.
• In this context, prealgebra trains us to solve problems step-by-step without skipping any foundational concepts.

By mastering prealgebra, students set themselves up for success in algebra and beyond. It provides the techniques needed to approach problems and develop logical reasoning skills.

Isolate Variable

Isolating the variable is a core task when solving equations. It involves getting the variable on one side of the equation by itself, which allows us to determine its value. In the given exercise:

• Our goal was to solve for \(x\) by removing the fraction that was paired with it.
• We did this by multiplying both sides of the equation by \(-2\), as that is the reciprocal of \(-\frac{1}{2}\).
• This multiplication gets \(x\) alone on one side, making the solution straightforward: \(x = 8\).

Generally, isolating variables involves reversing the operations surrounding the variable, ensuring a clear path to uncovering its value. This empowers students to tackle more complex multistep equations with confidence as they advance in their studies.