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Chapter 2: Problem 34

Solve each equation, and check your solution. $$ 6(2 x+8)=4(3 x-6) $$

Short Answer

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Step by step solution

01

Distribute the constants

Distribute the constants on each side of the equation: On the left side: 6(2x + 8) = 12x + 48 On the right side: 4(3x - 6) = 12x - 24
02

Write the new equation

After distribution, the equation becomes: 12x + 48 = 12x - 24
03

Subtract 12x from both sides

In order to simplify, subtract 12x from both sides of the equation: 12x + 48 - 12x = 12x - 24 - 12x This simplifies to 48 = -24
04

Analyze the result

Notice that the resulting statement, 48 = -24, is false. This indicates there is no solution for the equation.
05

Conclusion

Since we ended up with a false statement, the original equation has no solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is an essential concept in algebra. It allows us to multiply a single term across each term within a parenthesis. This is how we distribute the constants in the equation. For example, in the equation given as:
$$6(2x + 8) = 4(3x - 6)$$
applying the distributive property:
  • On the left side: \(6 \times (2x + 8)\). This means we multiply 6 by 2x and 6 by 8, which gives us \(12x + 48\).
  • On the right side: \(4 \times (3x - 6)\). Similarly, we multiply 4 by 3x and 4 by -6 to get \(12x - 24\).

Understanding the distributive property simplifies complex algebraic expressions, making solving equations more straightforward.
Simplifying Equations
After using the distributive property, the next step is to simplify the equation. Simplification involves combining like terms and moving terms to one side of the equation. Consider our distributed equation:
$$12x + 48 = 12x - 24$$
The goal is to isolate the variable.
  • Start by subtracting \(12x\) from both sides: \(12x + 48 - 12x = 12x - 24 - 12x\). This simplifies the equation to \(48 = -24\).

Simplification helps in recognizing the nature of the equation. If we can isolate the variable successfully, we can find the solution.
No Solution Equations
Sometimes, after simplifying an equation, we end up with a statement that is inherently false, such as \(48 = -24\). This indicates that there are no values that can satisfy the equation. In our case, following simplification, we reached:
$$48 = -24$$
  • A false statement like this means there is no possible value for \(x\) that can make the equation true.
  • This conclusion leads us to state that the original equation has no solutions.

Recognizing an equation with no solution is crucial, as it tells us that the equation does not represent a possible scenario within the given context. Always double-check your work to ensure accuracy.

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