Question

amaya used these steps to solve the equation ( 8x + 4 = 9 + 4(2x - 1) ). which choice describes the meaning of her result, ( 4 = 5 )?

amaya made a mistake because 4 is not equal to 5.

no values of ( x ) make the equation true.

all values of ( x ) make the equation true.

the solution is ( x = 4 ) or 5.

Explanation:

Step1: Analyze the equation solving process

When solving an equation, if we end up with a false statement like \(4 = 5\) (a contradiction), it means there are no values of the variable (\(x\) in this case) that can satisfy the original equation.

Step2: Evaluate each option

• Option "Amaya made a mistake because \(4\) is not equal to \(5\)": The steps of expanding \(4(2x - 1)=8x - 4\) and subtracting \(8x\) from both sides are correct algebraic operations. It's not a mistake in calculation, but rather an indication of the nature of the solution.
• Option "No values of \(x\) make the equation true": Since \(4 = 5\) is a contradiction, there are no \(x\) values that can make the original equation \(8x + 4=9 + 4(2x - 1)\) hold true.
• Option "All values of \(x\) make the equation true": This would be the case if we ended up with a statement like \(0 = 0\) (an identity). But \(4 = 5\) is not an identity.
• Option "The solution is \(x = 4\) or \(5\)": Substituting \(x = 4\) into the original equation: \(8\times4+4=32 + 4=36\), and \(9+4(2\times4 - 1)=9+4\times7=9 + 28 = 37\). Substituting \(x = 5\): \(8\times5+4=40+4 = 44\), and \(9+4(2\times5 - 1)=9+4\times9=9+36 = 45\). So \(x = 4\) and \(x = 5\) do not satisfy the equation.

Answer:

No values of \(x\) make the equation true.