Question

which statement represents the simplified form of the given equation and correctly describes the solution?
$9x - 4 = 2(8 + 4x) + 7$
a. $x = 27$; exactly one real solution
b. $16 \
eq 11$; no real solution
c. $x = 19$; exactly one real solution

Explanation:

Step1: Expand the right - hand side

First, we expand the expression \(2(8 + 4x)\) using the distributive property \(a(b + c)=ab+ac\). So, \(2(8 + 4x)=2\times8+2\times4x = 16+8x\). The original equation \(9x - 4=2(8 + 4x)+7\) becomes \(9x-4 = 16 + 8x+7\).

Step2: Simplify the right - hand side

Combine like terms on the right - hand side. \(16 + 7=23\), so the equation is \(9x-4=8x + 23\).

Step3: Solve for x

Subtract \(8x\) from both sides of the equation: \(9x-8x-4=8x - 8x+23\), which simplifies to \(x-4 = 23\). Then add 4 to both sides: \(x-4 + 4=23 + 4\), so \(x = 27\). Since we have found a unique value for \(x\), there is exactly one real solution.

Wait, there is a mistake in the above steps. Let's re - do it:

Step1: Expand the right - hand side

Using the distributive property \(a(b + c)=ab + ac\), for \(2(8 + 4x)\), we get \(2\times8+2\times4x=16 + 8x\). The equation \(9x-4 = 2(8 + 4x)+7\) becomes \(9x-4=16 + 8x+7\).

Step2: Combine like terms on the right

\(16 + 7 = 23\), so the equation is \(9x-4=8x + 23\).

Step3: Isolate the x - term

Subtract \(8x\) from both sides: \(9x-8x-4=8x-8x + 23\), which gives \(x-4=23\).

Step4: Solve for x

Add 4 to both sides: \(x-4 + 4=23 + 4\), so \(x = 27\). So the equation has exactly one real solution, and \(x = 27\).

Answer:

A. \(x = 27\); exactly one real solution