Problem 85 - Elementary Algebra

Question

Solve each of the following equations. Don't forget that division by zero is undefined. $$\frac{x-2}{2}=\frac{x}{2}-1$$

Step-by-Step Solution

Answer

Infinitely many solutions.

Step 1: Understand the Equation

The given equation is \(\frac{x-2}{2}=\frac{x}{2}-1\). Both sides are fractions, so we need to find \(x\) that makes this balance, keeping in mind that division by zero is undefined.

Step 2: Clear the Fractions by Finding Common Terms

Multiply each term by 2 to eliminate the denominators. This gives us:\[2 \times \frac{x-2}{2} = 2 \times \left(\frac{x}{2} - 1\right)\]Simplifying, we get:\[ x-2 = x - 2\]

Step 3: Solve for x

Now, we have the equation \(x-2 = x-2\). This is a true statement for any value of \(x\), which indicates that there are infinitely many solutions, as long as the operation is valid.

Step 4: Check for Division by Zero

Review the steps to ensure no step involves dividing by zero. Since the denominator was clear from the start and multiplied out, we don’t need to worry about division by zero in the simplified equation.

Key Concepts

Elementary Algebra

Elementary algebra forms the foundation of algebraic learning. It's the area of mathematics where we learn to manipulate expressions and solve equations that involve unknowns or variables, like \(x\). One of the first steps in solving equations, as demonstrated in the original exercise, is to aim for equivalence by performing operations that simplify the equation. These can include eliminating fractions by multiplication, adding or subtracting terms on both sides, and isolating variables by division or multiplication.

When working with fractions, as seen in the exercise, multiplying each term by a common denominator helps clear the fraction and simplifies the equation. This practice is fundamental and serves to make equations easier to handle. In this step, both sides of the equation are balanced, meaning that any operation done on one side must also be done on the other. This balance maintains the equation’s integrity and allows you to find the value of the unknown variable effectively.

Infinite Solutions

In mathematics, specifically in solving equations, the term 'infinite solutions' describes a scenario where any value of the variable will satisfy the equation. This is what happens in equations that are true for all values of the unknown, like the equation we solved above: \(x-2 = x-2\).

• This tells us that both sides of the equation are identical and only differ due to equivalent operations made during the solving process.
• Whenever both sides of an equation simplify to the same expression, this means there are infinitely many solutions, simply because they will always balance irrespective of the specific value of the variable.

Even though infinite solutions can seem confounding at first, they can be identified by the point where no further simplification or isolation of the variable results in a unique solution. Understanding infinite solutions is crucial because it highlights that not every algebraic problem has a singular, neat solution.

Division by Zero

Division by zero is a fundamental concept in mathematics that always leads to undefined results. When an equation involves division by zero, it ceases to be valid because it's mathematically impossible to divide a number into zero parts. Thus, mathematicians always caution against letting the denominator of a fraction become zero during calculations.

In the example provided, you initially have fractions but immediately multiply to clear the fractions; thereby eliminating the possibility of dividing by zero. It's essential to ensure the operations performed do not inadvertently introduce division by zero.

• Always check the denominator before proceeding with division.
• If the denominator can become zero, explore alternative methods to solve the equation or constrain the domain of the function involved.

By keeping these principles in mind and taking care to avoid such scenarios, you maintain mathematical correctness throughout your solving process.