Question

determine whether each equation is an identity, a conditional equation, or a contradiction give the solution set.
5x - 3 = 2(2x + 2) - 1
is the equation a conditional equation, an identity, or a contradiction?
a identity
b conditional equation
c contradiction
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 the right - hand side

We start with the equation \(5x - 3=2(2x + 2)-1\). First, we use the distributive property \(a(b + c)=ab+ac\) to expand the right - hand side. So, \(2(2x + 2)=2\times2x+2\times2 = 4x + 4\). Then the equation becomes \(5x-3 = 4x + 4-1\).

Step2: Simplify the right - hand side

Simplify \(4x + 4-1\) to get \(4x+3\). So now our equation is \(5x - 3=4x + 3\).

Step3: Solve for x

Subtract \(4x\) from both sides of the equation: \(5x-4x-3=4x - 4x+3\), which simplifies to \(x - 3=3\). Then add 3 to both sides: \(x-3 + 3=3 + 3\), so \(x = 6\).

Since the equation is true only for \(x = 6\) (not for all real numbers and not for no real numbers), it is a conditional equation and the solution set is \(\{6\}\).

Answer:

The equation is a conditional equation. The solution set is \(\{6\}\), so we choose option A and fill in 6. So the answer for the first part (type of equation) is B. conditional equation, and for the solution set, we choose A with the solution set \(\{6\}\).