Problem 41 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(-2(x+3)-x-4=-3(x+4)+2\)

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Distribute the constants

First, distribute the -2 on the left side and -3 on the right side of the equation: -2(x + 3) - x - 4 = -3(x + 4) + 2 Becomes: -2x - 6 - x - 4 = -3x - 12 + 2

Step 2: Combine like terms on both sides

Combine the like terms on both sides of the equation: On the left side: -2x - x - 6 - 4 = -3x - 10 => -3x - 10 On the right side: -3x - 12 + 2 = -3x - 10

Step 3: Compare both sides

After simplification both sides of the equation: -3x - 10 = -3x - 10

Step 4: Determine type of equation

Since both sides of the equation are identical, the equation is an identity.

Key Concepts

Distributive Property

The distributive property is a core concept in algebra used to simplify expressions and solve equations. It states that multiplying a number by a sum is the same as multiplying the number by each addend individually and then summing the products. For example: a(b + c) = ab + ac.In the given problem -2(x + 3) - x - 4 = -3(x + 4) + 2,we apply the distributive property to -2(x + 3) on the left side and -3(x + 4) on the right side. This gives us:-2x - 6 - x - 4 = -3x - 12 + 2. Using the distributive property simplifies complex equations and makes it easier to combine like terms.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions. Like terms are terms that have the same variable raised to the same power. For example, in the expression -2x - 6 - x - 4, -2x and -x are like terms because they both contain the variable x. Similarly, -6 and -4 are like terms because they are both constants. By combining these like terms, we get: -2x - x = -3x and -6 - 4 = -10. So, the expression simplifies to -3x - 10.This step is crucial for simplifying equations and solving them more efficiently.

Identity Equation

An identity equation is an equation that is true for all values of the variable. These equations often result in an expression where both sides are identical after simplification. In our exercise, after simplifying both sides of the equation, we get: -3x - 10 = -3x - 10.Since both sides are the same, this means the original equation is an identity. No matter what value we substitute for x, the equation will always hold true. Recognizing identity equations can help us understand the nature of the solutions for a given problem.

College Algebra

College algebra involves understanding and applying various algebraic concepts and techniques to solve equations and problems. It covers topics like the distributive property, combining like terms, and understanding different types of equations including identities and contradictions. Mastering these fundamentals helps in tackling more complex problems and is essential for higher-level math courses. Working through examples, like the one provided, helps in reinforcing these concepts and improving problem-solving skills, which are critical in college algebra.