Problem 61 - Introductory Algebra for College Students

Question

Solve each 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 Equation

Expand the left side of the equation \(2(x + 4)\) to get \(2x + 8\). Simplify the terms on the right side \(4x + 5 - 2x + 3\) to get \(2x + 8\). So, the equation becomes \(2x + 8 = 2x + 8\)

Step 2: Attempt to Solve

Subtract \(2x\) from both sides of the equation, this results in \(8 = 8\).

Step 3: Analyze the Result

After simplifying, \(8 = 8\) is a statement that is always true, regardless of the value of 'x'. Therefore the initial equation is true for all real numbers.

Key Concepts

Solving Equations

Solving equations forms the foundation of algebra. It's like unlocking a puzzle where we find the value of the variable that satisfies the given equation. In essence, it involves performing operations to isolate the variable on one side of the equation until we find its value. Here's a simplified approach:

• Start by simplifying both sides of the equation: combine like terms and eliminate any parenthesis by distributing.
• Use inverse operations to get the variable by itself on one side. This often involves adding, subtracting, multiplying, or dividing both sides of the equation.
• Check your solution by substituting the found value back into the original equation to ensure both sides are equal.

This process allows us to confront more complex equations systematically and ensures that any value we find actually fulfills the original conditions of the problem.

No Solution Equations

Sometimes, while solving equations, you might find that there is no possible value for the variable that makes the equation true. This happens when simplifying the equation leads to a contradiction.A common example of this is when you get a statement like \(3 = 5\), which is clearly never true. Such equations are called 'no solution' equations because there isn't any number that can replace the variable to make both sides of the equation equal.

• Be vigilant as you simplify the equation.
• If the variable terms disappear and you're left with a false statement, declare the equation has no solution.
• Consider that no solution doesn't imply the problem is unsolvable, but rather that no real number satisfies it.

True for All Numbers

An intriguing type of equation occurs when simplifying leads to a statement that is always correct, like \(8 = 8\). This tells us that the equation is true for all numbers. Essentially, any value of the variable will satisfy the equation.Such equations occur when both sides balance perfectly after simplifying, with all variable terms canceling out, leading to a true number statement. Here are some features of these equations:

• When simplifying, if you end up with an identity, such as \(a = a\), the equation is true for all real numbers.
• This indicates a dependent relationship where every value of the variable works.
• Recognizing these kinds of equations quickly simplifies your workload as you don’t need to seek specific solutions.

This means you have a flexible situation where there isn't just a single solution but an infinite number of them -- every number you substitute for the variable will make the original equation true.