Problem 66 - 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. \(5 x-5=3 x-7+2(x+1)\)

Step-by-Step Solution

Answer

The given equation is true for all real numbers.

Step 1: Simplify the right side

Expand and merge like terms on the right side of the equation. 3x - 7 + 2(x + 1) = 3x - 7 + 2x + 2, which simplifies to 5x - 5.

Step 2: Compare both sides of the equation

Now compare sides after the simplification. The equation now reads 5x - 5 = 5x - 5.

Step 3: Analyze the simplified equation

Since both sides of the equation are immutable, the equation is true for all real numbers.

Key Concepts

Equation Solving

Equation solving is a process in mathematics that involves finding the values of the variables that satisfy the equation. When you encounter an equation like \(5x - 5 = 3x - 7 + 2(x + 1)\), your goal is to determine the values of \(x\) that make the equation true. This often involves manipulating the equation to isolate the variable. Begin by simplifying or rewriting parts of the equation to make it easier to solve. In our example, you would first need to distribute the \(2\) through the expression \((x + 1)\) and then combine like terms.

Key steps include:

• Distributing multiplication over addition or subtraction.
• Combining like terms.
• Transforming the equation to a simpler form.

The final form of this equation, after simplification, is \(5x - 5 = 5x - 5\). This indicates a special scenario where further analysis is required to determine the solution set.

Algebraic Manipulation

Algebraic manipulation is an essential skill in solving equations and involves rearranging and simplifying expressions to isolate variables. The equation solving process uses various manipulation techniques, such as expanding expressions and combining like terms.

In our example, we expanded on the right side by distributing the \(2\) in \(2(x + 1)\), resulting in \(2x + 2\). This allows us to consolidate terms, leading us to \(5x - 5 = 5x - 5\).

• Expanding: Apply the distributive property, \(a(b + c) = ab + ac\), to remove parentheses.
• Combining like terms: Simplify expressions by adding or subtracting coefficients of similar terms, such as combining \(3x\) and \(2x\).
• Balancing equations: Ensure that the manipulation maintains equality on both sides of the equation.

Each of these steps is crucial to solving equations accurately and efficiently.

Real Numbers

In mathematics, real numbers include all the numbers on the number line. They comprise both rational and irrational numbers, and are essential in defining solutions to equations. When an equation reads like \(5x - 5 = 5x - 5\), it suggests a situation where every real number can be a solution.

In the realm of real numbers, every possible value of \(x\) satisfies the equality. Therefore, this kind of equation is considered an identity, which implies the equation is consistently true. This contrasts with equations that have:

• One unique solution, meaning only a specific real number satisfies it.
• No solution, such as in contradictions where no real number can fulfill the equation.

Recognizing when an equation holds for all real numbers is just as important as finding solutions for equations with specific or no solutions.