Problem 61 - 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. \(2(x+4)=4 x+5-2 x+3\)

Step-by-Step Solution

Answer

The equation is true for all real numbers.

Step 1 - Simplify the left hand side

First simplify the left hand side of the equation. Distribute the 2 on the left side of the equation to both x and 4: \(2(x+4) = 2x + 8\). Our equation becomes: \(2x + 8 = 4x + 5 - 2x + 3\).

Step 2 - Simplify the right hand side

Now let's simplify the right hand side of the equation by adding similar terms: \(4x - 2x + 5+3 = 2x + 8\). Our equation now becomes: \(2x + 8 = 2x + 8\).

Step 3 - Check if the equation is valid for all real numbers

Looking at the equation \(2x + 8 = 2x + 8\), we can see that it is equal for all values of x. We can conclude therefore that the given equation is true for all real numbers.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, operators, variables, and grouping symbols. For instance, in the exercise given, the expression on the left side of the equation, \(2(x+4)\), is an example of an algebraic expression.

To simplify an algebraic expression, you distribute any numbers outside the grouping symbol to each term inside the group. This was demonstrated in Step 1 of the solution when \(2\) was distributed to both \(x\) and \(4\), resulting in \(2x + 8\). Simplifying algebraic expressions helps to transform them into a simpler form which makes it easier to work with, especially when solving equations.

It's important to recognize like terms as well, which are terms that have the same variable raised to the same power. These can be combined to further simplify the expression. In our example, \(4x\) and \(–2x\) on the right-hand side of the equation are like terms, and simplifying them leads to \(2x\).

Equation Simplification

Equation simplification involves reducing an equation to its simplest form while maintaining its equality. In the context of solving linear equations, this often means combining like terms and eliminating unnecessary terms.

In Step 2, the solution process simplified the right-hand side of the equation by combining like terms. The operation \(4x - 2x + 5 + 3\) simplifies to \(2x + 8\), matching the already simplified left-hand side.

Simplification serves two purposes: it makes it easier to see the relationship between variables and constants, and it can often indicate the nature of the solution if one exists. In this case, after simplification, it became evident that the equation was true for all real numbers, meaning no specific solution exists as the equation is an identity - both sides are identical. This reveals that for every real number substituted for \(x\), the equation holds true.

Real Numbers

Real numbers encompass a wide range of values including integers, fractions, decimals, and irrational numbers. They are the most commonly used set of numbers in algebra and represent all possible values along the number line.

In Step 3 of the solution process, after simplifying both sides of the equation, it resolved into \(2x + 8 = 2x + 8\), indicating that no matter what real number value is substituted for \(x\), the equation will be true. This is an example of an identity in algebra where the equation does not just have one solution, but rather is valid for all real numbers.

Understanding the properties of real numbers is essential in algebra, as it helps in identifying the solution sets of equations. Real numbers obey the commutative, associative, and distributive properties, which are fundamental rules used in all simplification and equation solving processes.