Problem 97 - Intermediate Algebra

Question

Solve each equation. $$ 2(x-2)=\frac{2}{3}(3 x+8)-2 $$

Step-by-Step Solution

Answer

No solution exists for this equation.

Step 1: Distribute and simplify both sides

Start by distributing the constants on both sides of the equation. For the left-hand side, distribute 2 over \(x - 2\): \[ 2(x - 2) = 2x - 4 \]For the right-hand side, distribute \(\frac{2}{3}\) over \(3x + 8\): \[ \frac{2}{3}(3x + 8) = 2x + \frac{16}{3} \]Thus, the equation becomes: \[ 2x - 4 = 2x + \frac{16}{3} - 2 \]

Step 2: Combine like terms on the right side

Combine the constant terms on the right side of the equation. The current right side is \[ 2x + \frac{16}{3} - 2 \]First, express \(-2\) using the same denominator, 3: \[-2 = \frac{-6}{3}\]Then combine: \[ 2x + \frac{16}{3} - \frac{6}{3} = 2x + \frac{10}{3} \]

Step 3: Eliminate \(2x\) from both sides

Subtract \(2x\) from both sides of the equation to simplify:\[ 2x - 4 - 2x = 2x + \frac{10}{3} - 2x \]This simplifies to:\[ -4 = \frac{10}{3} \]

Step 4: Resolve the equation

The equation \(-4 = \frac{10}{3}\) seems incorrect in our process. This implies that there is a contradiction, meaning no solution exists for the equation because a constant does not equal another constant.

Key Concepts

Distributive Property

The distributive property is a core concept in algebra used to multiply a single term over two or more terms inside a parenthesis. It's essential for simplifying equations.

• To apply the distributive property, multiply the term outside the parenthesis by each term inside the parenthesis.
• This helps distribute the multiplication over addition or subtraction within the parenthesis.

In the original exercise, we applied the distributive property to both sides of the equation.
For the left-hand side, the term 2 is multiplied with both terms inside the parenthesis, which are

• 2 multiplied by x giving us: \[2x\]
• 2 multiplied by -2 giving us: \(-4\)

Thus, we derive \[ 2(x - 2) = 2x - 4 \]
On the right-hand side, the fraction \(\frac{2}{3}\) is distributed across

• 3x: \(\frac{2}{3} \times 3x = 2x\)
• 8: \(\frac{2}{3} \times 8 = \frac{16}{3}\)

Knowing how to effectively apply the distributive property simplifies equations and sets the foundation for solving them.

Simplifying Algebraic Expressions

Once you've distributed all necessary terms, simplifying the expression becomes key to making the equation manageable and ready for solving.

• Simplifying involves combining like terms. Like terms have the same variables raised to the same power.
• In this problem, on the right side, we have the terms \(2x\) and \(\frac{16}{3} - 2\).

To simplify, consider the constants first:

• Express \(-2\) as a fraction to ease the combination: \(-2 = \frac{-6}{3}\).
• Now, combine \(\frac{16}{3}\) and \(\frac{-6}{3}\): \(\frac{16}{3} - \frac{6}{3} = \frac{10}{3}\).

The right side now simplifies to \(2x + \frac{10}{3}\).
Simplifying expressions makes the process of solving the equation easier and cleaner.

Solving Equations with Fractions

Solving equations with fractions often requires careful manipulation to ensure a clear path to the solution.

• First, ensure fractions are combined properly. This often involves finding a common denominator.
• Later, the goal is to simplify the equation to isolate the variable and solve.

In our problem, after simplifying both sides, we subtract \(2x\) from both the sides as part of clearing the variable terms. This leads us to:
\[-4 = \frac{10}{3}\]

• This equation highlights a contradiction since two constants cannot equate if they are different.
• Such a result suggests no solution exists for the original equation.

Equations sometimes reveal inconsistencies, indicating a conceptual understanding of the possibility of no solutions in some algebraic equations.