Problem 67 - 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. \(4(x+2)+1=7 x-3(x-2)\)

Step-by-Step Solution

Answer

The equation has no solution, as \(9\) does not equal \(6\).

Step 1: Simplify the Equation

The given equation is \(4(x+2)+1=7x-3(x-2)\). Begin by applying the distributive property on both sides of the equation. This yields \(4x+8+1=7x-3x+6\), which simplifies further into \(4x+9=4x+6\).

Step 2: Solve for x

Now isolate the variable x by subtracting \(4x\) from both sides. This will provide the equation \(9=6\).

Step 3: Check for Possible Solutions

On simplifying, we find that \(9\) does not equal \(6\), which means the equation has no solution.

Key Concepts

No Solution Equation

In algebra, you may encounter equations that have no solution, meaning there's no possible value of the variable that can satisfy the equation. This happens when simplifying an equation leads you to a statement that is false, such as \(9 = 6\). In our exercise, we simplified \(4(x+2) + 1 = 7x - 3(x-2)\) to eventually arrive at \(9 = 6\). Since 9 is not equal to 6, there is no possible value of \(x\) that can make both sides of the original equation equal. If you ever reach such a conclusion, it's a clear sign that the equation has no solution. This means the variable doesn't have any value that works, hence it can be shown as an empty set \(\emptyset\) in set notation.

Distributive Property

The distributive property is a vital algebraic principle used to simplify expressions and solve equations. It states that \(a(b+c) = ab + ac\). This means you distribute the multiplication of \(a\) over each term within the parentheses. In the equation \(4(x+2)+1=7x-3(x-2)\), you use the distributive property twice:

• Multiply \(4\) by both \(x\) and \(2\), resulting in \(4x + 8\).
• Multiply \(-3\) by \(x\) and \(-2\), resulting in \(-3x + 6\).

After applying the distributive property correctly, the equation simplifies to \(4x + 9 = 4x + 6\). This factor-by-factor distribution lets you handle more complex equations by breaking them down into simpler components.

Solving Equations

Solving an equation involves finding all possible values of the variable that make the equation true. You do this by isolating the variable on one side of the equation. For the exercise's equation, you simplify it first: start with \(4(x+2)+1=7x-3(x-2)\). After applying the distributive property, reorganize like terms to get \(4x + 9 = 4x + 6\). Now, to isolate \(x\), try to remove \(4x\) from both sides. This leaves you with the statement \(9 = 6\), which indicates no solution, since it is false. Always check whether the simplified version of your equation leads to a meaningful statement that can help determine the potential values of a variable. If not, you may face a no solution scenario.

Set Notation

Set notation is a mathematical language used to describe sets, or collections, of numbers. In the context of solving equations, it's particularly useful when expressing solutions. For the equation in the exercise, no value of \(x\) satisfies it, resulting in no solution. In set notation, this is represented by an empty set, written as \(\emptyset\). On the other hand, when all real numbers are solutions, it might be expressed as \(\mathbb{R}\).

• An empty set indicates no solutions exist.
• The set of all real numbers \(\mathbb{R}\) means every number is a solution.

Using set notation provides a clear and concise way to express and communicate the solution set, or lack thereof, for any given equation.