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Chapter 2: Problem 36

Solve each equation, and check the solution. \(6 x-4(x+1)=2 x+4\)

Short Answer

Expert verified
No solution.

Step by step solution

01

- Distribute

Distribute the term in the equation. For the left-hand side, distribute -4 to both terms inside the parentheses.\[ 6x - 4(x + 1) = 2x + 4 \] \[ 6x - 4x - 4 = 2x + 4 \]
02

- Combine like terms

Combine the x-terms on the left-hand side of the equation.\[ (6x - 4x) - 4 = 2x + 4 \]\[ 2x - 4 = 2x + 4 \]
03

- Isolate the x-term

Subtract 2x from both sides of the equation to isolate the constant term.\[ 2x - 2x - 4 = 2x - 2x + 4 \]\[ -4 = 4 \]
04

- Check for consistency

Since \[ -4 eq 4 \], the equation is inconsistent and has no solution.

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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 algebraic property that we use to simplify expressions, especially those involving parentheses. It states that for any numbers a, b, and c, the equation a(b + c) = ab + ac holds true.
In the example equation, we start with:
\(6x - 4(x + 1) = 2x + 4\)
To eliminate the parentheses, we distribute \(-4\) into \((x + 1)\):
\[-4(x + 1) = -4x - 4 \]
Therefore, the equation becomes:
\(6x - 4x - 4 = 2x + 4\)
This step is crucial because it prepares the equation for further simplification.
Combining Like Terms
Combining like terms is another fundamental concept where terms with the same variable are added or subtracted to simplify an equation. In our example, after distributing, our equation is:
\(6x - 4x - 4 = 2x + 4\)
The next step is to combine the like terms (the terms with the variable x):
\(6x - 4x = 2x\)
This simplifies our equation to:
\(2x - 4 = 2x + 4\)
Combining like terms makes it easier to isolate the variable and solve the equation.
Always combine terms to reduce complexity before moving forward to the next steps.
Isolating Variables
Isolating variables means rearranging the equation so that the variable you're solving for is by itself on one side. In our simplified equation:
\(2x - 4 = 2x + 4\)
We subtract \(2x\) from both sides to isolate the constant terms:
\(2x - 2x - 4 = 2x - 2x + 4 \)
The result is:
\(-4 = 4\)
Notice that there are no \(x\) terms left on either side. The constants are isolated, but something seems off.
Inconsistent Equations
Inconsistent equations are equations that have no solutions. When you simplify an equation and end up with a false statement, such as:
\(-4 = 4\)
This tells us that our initial equation is inconsistent鈥攖here are no values for \(x\) that can satisfy the equation.
Inconsistent equations often occur when the variables cancel out, and you're left with a contradiction. It's important to recognize these cases, as they indicate that the original problem has no solution.

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