2(x + 3) = 6x + 2
Understand the Problem
The question is asking to solve the equation 2(x + 3) = 6x + 2 and determine whether the resulting statement is a conditional, contradiction, or identity. We will solve the equation to find out which of these options applies.
Answer
The equation is an identity.
Answer for screen readers
The equation is an identity.
Steps to Solve
- Distribute on the left side
Start by distributing the 2 into the parentheses:
$$ 2(x + 3) = 2x + 6 $$
This gives you the equation:
$$ 2x + 6 = 6x + 2 $$
- Rearrange the equation
Next, we will get all terms involving $x$ on one side and constant terms on the other. Subtract $2x$ from both sides:
$$ 6 = 6x - 2x + 2 $$
This simplifies to:
$$ 6 = 4x + 2 $$
- Isolate the variable
Now, subtract 2 from both sides:
$$ 6 - 2 = 4x $$
This simplifies to:
$$ 4 = 4x $$
- Solve for x
Divide both sides by 4:
$$ x = 1 $$
- Determine the type of statement
Substituting $x = 1$ back into the original equation to check if both sides are equal:
Left side:
$$ 2(1 + 3) = 2 \cdot 4 = 8 $$
Right side:
$$ 6(1) + 2 = 6 + 2 = 8 $$
Since both sides are equal (8 = 8), the equation holds true for this value, indicating that the equation is an identity.
The equation is an identity.
More Information
An identity means the equation holds true for all values of the variable. In this case, any $x$ that satisfies the equation will result in both sides being equal.
Tips
- Forgetting to distribute correctly when expanding.
- Not rearranging the equation properly can lead to incorrect solutions.
- Failing to check the solution by substituting it back into the original equation.
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