Problem 46 - Algebra for College Students

Question

Solve each equation, and check the solution. If applicable, tell whether the equation is an identity or a contradiction. \(-[6 x-(4 x+8)]=9+(6 x+3)\)

Step-by-Step Solution

Answer

The solution is x = -1/2 and it's neither an identity nor a contradiction.

Step 1: Distribute and Simplify

Distribute the negative sign on the left-hand side of the equation:

Step 2: Combine like terms

On the left-hand side, combine like terms inside the bracket: -[6x-(4x+8)] On simplifying inside the brackets, we get: -[6x-4x-8] Then on further distribution of the negative sign across, we get : -(2x-8)=9+(6x + 3) and then further simplify to get, -2x + 8=9 + 6x + 3

Step 3: Transfer variables to one side

Move the variable terms to one side of the equation: Combining constants on one side, we get, -2x + 8 = 6x + 12 and then move all the x's variable to one side, i.e. Adding 2x to both sides, we get, 8 - 12 = 6x + 2x + 12 which further results into, 8 - 12 = 8x finally resulting into : -4 = 8x

Step 4: Solve for x

Divide both sides by 8:

Step 5: Check the solution

Verify the solution by substituting the value of x back into the original equation: left hand side evaluation and then right-hand side evaluation respectively. This is to cross check the solution if both sides match the values gotten at different stages of working.

Key Concepts

distributive property

The Distributive Property is a fundamental principle in algebra that helps to simplify expressions. It states that a(b + c) = ab + ac.
When solving an equation like \(-[6x-(4x+8)]=9+(6x+3)\), we begin by distributing the negative sign across the terms in the parentheses. This means we need to apply the negative sign to each term inside the bracket:
Step-by-step, we do the following:

• \(6x - (4x + 8)\)
• Distribute the negative sign: \(6x - 4x - 8\)

Properly using the distributive property makes the equation easier to manage, paving the way for the next steps in solving it.

combining like terms

Combining Like Terms is a method used to simplify expressions and make equations more manageable. Like terms are terms that have the same variables raised to the same power.
In the equation, after distributing the negative sign, we have: \(-[6x-4x-8] = 9 + (6x + 3)\).
To further simplify, combine the terms that contain the variable x:

• \(6x - 4x\) results in \(2x\)

Thus, the equation simplifies to:
\(-(2x - 8) = 9 + (6x + 3))\), which further simplifies to \(-2x + 8 = 9 + 6x + 3\). Now, it’s much easier to proceed with solving the equation because we have simplified it by combining like terms.

variable isolation

Variable Isolation is the process of getting the variable on one side of the equation by itself. After simplifying, we had:
\(-2x + 8 = 9 + 6x + 3\).
The next step is to move all the variable terms to one side and the constants to the other. We can do this by using basic algebraic operations like addition or subtraction.
First, let’s combine the constants on the right side:

• The equation now looks like: \(-2x + 8 = 6x + 12\).
• Add \(2x\) to both sides to isolate the variable: \(8 = 8x + 12\).

To completely isolate the variable, subtract 12 from both sides:
\(8 - 12 = 8x\), which simplifies further to \(-4 = 8x\).
Now, the variable is almost by itself, and we can move to the final step of solving for x.

solving for x

Solving for x means finding the value of the variable that makes the equation true. We have the simplified equation:
\(-4 = 8x\).
To isolate x, divide both sides of the equation by 8:
\(-4 / 8 = x\),
which simplifies to:
\(x = -0.5\). To check our solution, substitute \(x = -0.5\) back into the original equation:

• Left side: \(-[6(-0.5)-(4(-0.5)+8)]\) simplifies to \(1 - 4 - 8\), and further to \(1 - 12 = -11\).
• Right side: \(9 + 6(-0.5) + 3\) simplifies to \(9 - 3 + 3\), which is \(9 - 3 + 3 = 9\).

Both sides of the equation are equal when \(x = -0.5\), verifying that our solution is correct and consistent.