Problem 68 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. \(5 x-3(x+1)=2(x+3)-5\)

Step-by-Step Solution

Answer

The equation has no solution. There's no real number value for \(x\) that can make both sides of the equation equal, as the equation simplifies to an untrue statement (-3 = 1).

Step 1: Expand the brackets

Begin by expanding brackets on the both sides of the equation. This gives us: \(5x - 3x - 3 = 2x + 6 - 5\) which simplifies to \(2x - 3 = 2x + 1\)

Step 2: Simplify both sides and solve for x.

To isolate \(x\) on one side, let's subtract \(2x\) from both sides of the equation: \(-3 = 1\). This represents an untrue statement.

Step 3: State the outcome

An untrue equation points to there being no value of \(x\) that makes both sides of the equation equal. As such, the equation has no solution.

Key Concepts

No Solution Equations

Understanding equations with no solution is critical for mastering algebra. Equations that have no solution are often identified by simplifying the equation to the point where you end up with a statement that is always untrue, no matter what value is substituted for the variable. In our exercise, the simplification process led to equation that the solution for equation has no solution. This characteristic is commonly found in equations where the variable terms cancel out, leaving a false statement, such as equation representing an impossibility in real number terms. Recognizing 'no solution' scenarios is as important as solving for an actual value of x because it signifies that the equation is inconsistent with real numbers. In the course of studying algebra, you will come across statements that might not seem like they have a solution initially. However, by applying equation simplification techniques, you can reveal the true nature of these equations.

Equation Simplification

Simplifying algebraic equations is perhaps the most fundamental skill one must develop when learning algebra. The goal of simplification is to make the equation easier to understand and solve, often by combining like terms and reducing the equation to its simplest form. In our example, the process includes expanding brackets and then combining the terms that contain the variable, as well as combining constant terms to end up with a format where the variable is on one side of the equals sign, and the constants are on the other.

When all like terms are combined (for instance, combining all the terms with 'x' in our example), and the equation is still not true for any real number, this signifies that the equation has no solution. Simplifying helps to clearly show the relationship—or lack thereof—between the variables and constants within the equation.

Expanding Brackets

Expanding brackets is a critical process in algebra that involves multiplying everything inside the bracket by the term that is outside the bracket. This is often one of the first steps to solve an equation because it allows you to transform an equation into a simpler form without brackets, making it easier to see the structure of the algebraic expression and combine like terms.

The order of operations dictates that you handle brackets before dealing with addition or subtraction. In the exercise provided, expanding brackets is the starting move that changes the equation's appearance. This step is essential to move on to the simplification process. It's important to ensure all terms within brackets are multiplied by the term outside to avoid any mistakes that could lead to incorrect conclusions regarding the nature of the solution.