Problem 143 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I need to be able to perform operations with fractions to determine whether \(\frac{3}{2}\) is a solution of \(8 x=12\left(x-\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

\(\frac{3}{2}\) is not a solution to the given equation.

Step 1: Substitute the value

Replace x in the equation with \(\frac{3}{2}\). So, the equation becomes: \(8 * \frac{3}{2} = 12 *\left(\frac{1}{2}\right)\)

Step 2: Perform the operations

Perform the multiplication on both sides of the equation to simplify it. This gives: \(12 = 12 * \frac{1}{2}\)

Step 3: Simplify the equation

Continuing with the operations on the right-hand side, we get: \(12 = 6\)

Step 4: Verify the equality

As 12 is not equal to 6, \(\frac{3}{2}\) doesn't satisfy the equation. Therefore, \(\frac{3}{2}\) is not a solution to the given equation.

Key Concepts

Operations with Fractions

Understanding operations with fractions is fundamental when solving algebraic equations that involve fractional components. When we deal with fractions, there are a few key operations we might perform: addition, subtraction, multiplication, and division. To multiply fractions, for instance, we multiply the numerators (top numbers) together and the denominators (bottom numbers) together. In the context of our exercise, where we multiply \(8\) by \(\frac{3}{2}\), we perform the multiplication as follows: \(8 \times \frac{3}{2} = \frac{8 \times 3}{2}\). This results in \(12\) when we simplify it as \(\frac{24}{2}\).

It's also important to simplify fractions whenever possible by dividing the numerator and denominator by their greatest common divisor. Simplification can make other operations easier and help us to clearly see whether a fraction can be a solution to the equation.

Substitution Method

The substitution method is an essential technique for solving equations, where you replace a variable with its value to simplify the equation and determine if the value is indeed a solution. In our case, we are testing whether \(\frac{3}{2}\) is a solution to the equation \(8x = 12\left(x-\frac{1}{2}\right)\). By substituting \(x\) with \(\frac{3}{2}\), the equation takes on a form that allows us to perform arithmetic operations to verify the proposed solution. Every time we substitute a value, it's key to replace every instance of the variable with the value to ensure accuracy.

Correct substitution leads to an equation with numbers and no variables, which can be straightforwardly evaluated to determine whether the substitution leads to a true statement.

Equation Simplification

Equation simplification involves reducing an equation to its simplest form, making it easier to solve or evaluate. This often includes combining like terms, simplifying expressions with fractions, and performing multiplication or division as necessary. In our exercise, after substitution, the equation simplification shows us that the right-hand side, originally \(12\left(\frac{3}{2}-\frac{1}{2}\right)\), transforms into \(12 \times 1\), which simplifies to \(12\).

Always keeping track of simplifications step by step is vital to maintain accuracy and to avoid common mistakes, like overlooking a negative sign or incorrectly simplifying an expression. It's the these small details that often determine whether your final answer is correct.

Solution Verification

Once an alleged solution to an equation has been substituted and the resulting expression simplified, it's time for solution verification. This step is about checking if the simplified, substituted equation holds true. In the provided problem, after simplification, we ended with \(12 = 6\). If both sides of the equation are equal after substituting the proposed solution, then the solution is valid. However, in this case, because \(12 \eq 6\), we conclude that \(\frac{3}{2}\) is not a true solution to the original equation.

Verification is essential because it confirms whether the original problem statement is satisfied. It's the final step that either validates the prior work or signals an error in earlier steps. Without this step, we cannot be sure of our answer's correctness.