Question

question
determine whether the equation below has one solution, no solutions, or an infinite number of solutions. afterwards, determine two values of ( x ) that support your conclusion.

( x - 3 = 3 - x )

answer attempt 1 out of 2
the equation has one solution (downarrow).
a value of ( x ) that makes the equation true is (square). when substituted into the equation and fully evaluated, it turns the equation into (square = square).
a value of ( x ) that makes the equation false is (square). when substituted into the equation and fully evaluated, it turns the equation into (square = square).

Explanation:

Step1: Solve the equation \(x - 3 = 3 - x\)

Add \(x\) to both sides: \(x + x - 3 = 3 - x + x\) → \(2x - 3 = 3\)
Add 3 to both sides: \(2x - 3 + 3 = 3 + 3\) → \(2x = 6\)
Divide by 2: \(x = 3\)
So the equation has one solution (\(x = 3\)).

Step2: Find a value that makes it true (use \(x = 3\))

Substitute \(x = 3\) into \(x - 3 = 3 - x\):
Left side: \(3 - 3 = 0\)
Right side: \(3 - 3 = 0\)
So \(0 = 0\).

Step3: Find a value that makes it false (use \(x = 0\))

Substitute \(x = 0\) into \(x - 3 = 3 - x\):
Left side: \(0 - 3 = -3\)
Right side: \(3 - 0 = 3\)
So \(-3 = 3\) (which is false).

Answer:

The equation has one solution.
A value of \(x\) that makes the equation true is \(3\). When substituted into the equation and fully evaluated, it turns the equation into \(0 = 0\).
A value of \(x\) that makes the equation false is \(0\). When substituted into the equation and fully evaluated, it turns the equation into \(-3 = 3\).

(Note: The false value can be any number except 3, e.g., \(x = 1\) would give \(1 - 3 = -2\) and \(3 - 1 = 2\), so \(-2 = 2\), also valid.)