Problem 59 - 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. $$3 x-7=3(x+1)$$

Step-by-Step Solution

Answer

The equation \(3x - 7 = 3(x + 1)\) has no solution.

Step 1: Simplify and isolate terms

First, distribute the 3 on the right side of the equation through the parenthesis, which results in \(3x - 7 = 3x + 3\). Now, subtract \(3x\) from both sides to balance the equation, which leaves \(-7 = 3\).

Step 2: Identify the nature of the solution

The equation \(-7 = 3\) is not possible because \(-7\) cannot be equal to \(3\), hence this equation has no solution.

Key Concepts

No Solution

When an equation is said to have "no solution," it means there are no values of the variable that can satisfy the equation. In simple words, no number exists that will make the equation true.

In our example, after simplifying the equation, we ended up with \[-7 = 3\]. This expression is false because \(-7\) can never be equal to \(3\).

Here's a bit of advice for recognizing when an equation has no solution:

• After simplifying, if you get an equation where the numbers are unequal like \(a = b\), where \(a\) and \(b\) are different constants, then the equation has no solution.
• It often results when the variable terms on both sides cancel out completely and you're left with a false statement.
• In words, you might express this by saying "there are no values of \(x\) that make the equation true."

Real Numbers

Real numbers are a crucial part of mathematics and include all the numbers that we usually think of, including integers, fractions, and decimals.

To better understand, here are some key points about real numbers:

• They include both rational numbers (like 7 or -1.5) and irrational numbers (like \(\pi\) or \(\sqrt{2}\)).
• Real numbers cover everything on the number line without any gaps.
• In the context of solving equations, if an equation is true for all real numbers, it means any real number can make the equation valid.

In our example equation \(3x - 7 = 3(x + 1)\), we know it does not hold for any real number as it results in an impossible statement, indicating no solution.

Distributive Property

The distributive property is a fundamental algebra concept that helps in multiplying a single term over terms inside a parenthesis.

Take a look at how it works:

• Mathematically, it is expressed as \(a(b + c) = ab + ac\). You distribute the outer term over each term inside the parenthesis.
• In our problem, we used the distributive property when simplifying \(3(x + 1)\). This results in \(3x + 3\).
• It's vital in simplifying equations and is often one of the first steps in solving algebraic expressions.

Understanding the distributive property allows you to break down equations into manageable parts, making it easier to solve or to identify when an equation has no solution.