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Decide whether the equation is conditional, an identity, or a contradiction. Give the solution set.

[tex]\[ 2x + 2(x - 4) = 5x + 1 \][/tex]

Is the equation a conditional, an identity, or a contradiction?

A. Conditional
B. Contradiction
C. Identity

Sagot :

GinnyAnswer
Sure, let’s tackle the problem step-by-step.

Step 1: Write down the given equation
[tex]\( 2x + 2(x - 4) = 5x + 1 \)[/tex]

Step 2: Simplify the left-hand side (LHS) of the equation
First, distribute the 2 into the terms inside the parenthesis:
[tex]\[ 2x + 2(x - 4) = 2x + 2x - 8 \][/tex]

Combine like terms on the LHS:
[tex]\[ 2x + 2x - 8 = 4x - 8 \][/tex]

So, the simplified equation is:
[tex]\[ 4x - 8 = 5x + 1 \][/tex]

Step 3: Bring all terms involving [tex]\(x\)[/tex] to one side and constants to the other side
Subtract [tex]\(4x\)[/tex] from both sides:
[tex]\[ 4x - 8 - 4x = 5x + 1 - 4x \][/tex]
[tex]\[ -8 = x + 1 \][/tex]

Now, subtract 1 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ -8 - 1 = x \][/tex]
[tex]\[ -9 = x \][/tex]

Thus, the solution to the equation is:
[tex]\[ x = -9 \][/tex]

Step 4: Categorize the equation
We need to determine whether the equation is:
- Conditional:
- An equation that is true for one or more, but not all, values of the variable.
- An identity:
- An equation that is true for all values of the variable.
- A contradiction:
- An equation that is never true, for any value of the variable.

Since the equation simplifies to a single specific value [tex]\(x = -9\)[/tex], it is true for this specific value only. This means the equation is conditional.

Conclusion
- Type of Equation: Conditional
- Solution Set: [tex]\(\{ -9 \}\)[/tex]

Therefore, the equation [tex]\(2x + 2(x - 4) = 5x + 1\)[/tex] is a conditional equation, and the solution set is [tex]\(\{ -9 \}\)[/tex].
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