Question

determine whether the equation is a conditional equation, an identity, or a contradiction. give the solution set. 10(v + 2) - 4v = 2(3v + 2) - 8
is the equation a conditional equation, an identity, or a contradiction?
a. contradiction
b. identity
c. conditional equation
select the correct choice below and fill in any answer boxes present in your choice.
a. the solution set is
(type an integer or a fraction. use a comma to separate answers as needed.)
b. the solution is all real numbers.
c. the solution is the empty set.

Explanation:

Step1: Expand both sides

Left: $10(v+2)-4v = 10v + 20 - 4v = 6v + 20$
Right: $2(3v+2)-8 = 6v + 4 - 8 = 6v - 4$
Correction: Re-calculate right side
Right: $2(3v+2)-8 = 6v + 4 - 8 = 6v - 4$
Re-evaluate left side correctly
Left: $10(v+2)-4v = 10v + 20 - 4v = 6v + 20$
Wait, correction:

Step1: Expand left-hand side

$10(v+2)-4v = 10v + 20 - 4v = 6v + 20$

Step2: Expand right-hand side

$2(3v+2)-8 = 6v + 4 - 8 = 6v - 4$

Step3: Simplify to compare

Set equal: $6v + 20 = 6v - 4$
Subtract $6v$: $20 = -4$
Error correction: Recheck original equation
Original equation: $10(v+2)-4v = 2(3v+2)-8$
Recompute left: $10v + 20 - 4v = 6v + 20$
Recompute right: $6v + 4 - 8 = 6v - 4$
Wait, no—wait, correct right side: $2(3v+2)-8 = 6v + 4 - 8 = 6v - 4$
Wait, this gives $6v+20=6v-4$ → $20=-4$, which is false. But this is a contradiction? No, wait, recheck the problem:
Wait, maybe a typo? No, let's redo:

Step1: Expand left expression

$10(v+2)-4v = 10v + 20 - 4v = 6v + 20$

Step2: Expand right expression

$2(3v+2)-8 = 6v + 4 - 8 = 6v - 4$

Step3: Set equal and simplify

$6v + 20 = 6v - 4$
Subtract $6v$ from both sides: $20 = -4$
This is a false statement for all $v$, so it is a contradiction, solution set is empty.

Corrected Answer:

A. contradiction
C. The solution is the empty set.

Step1: Expand left-hand side

$10(v+2)-4v = 10v + 20 - 4v = 6v + 20$

Step2: Expand right-hand side

$2(3v+2)-8 = 6v + 4 - 8 = 6v - 4$

Step3: Simplify the equation

Subtract $6v$ from both sides:
$6v + 20 - 6v = 6v - 4 - 6v$
$20 = -4$

Step4: Classify the equation

The statement $20=-4$ is always false, so the equation is a contradiction with no solutions (empty set).

Answer:

B. identity
B. The solution is all real numbers.