Question

solve for all values of b in simplest form. |-2b + 2| = 13 answer attempt 2 out of 2 additional solution remove solution b = -11/2, b = -15/2 submit answer

Explanation:

Step1: Apply absolute value definition

The equation \(|-2b + 2| = 13\) means \(-2b + 2 = 13\) or \(-2b + 2 = -13\) (since \(|x| = a\) implies \(x = a\) or \(x = -a\) for \(a\geq0\)).

Step2: Solve first equation \(-2b + 2 = 13\)

Subtract 2 from both sides: \(-2b = 13 - 2 = 11\).
Divide by -2: \(b = \frac{11}{-2}=-\frac{11}{2}\).

Step3: Solve second equation \(-2b + 2 = -13\)

Subtract 2 from both sides: \(-2b = -13 - 2 = -15\).
Divide by -2: \(b = \frac{-15}{-2}=\frac{15}{2}\)? Wait, no—wait, original second equation solution in the image was \(b = -\frac{15}{2}\)? Wait, no, let's re - check. Wait, \(-2b=-15\), so \(b=\frac{-15}{-2}=\frac{15}{2}\)? But the image has \(b = -\frac{15}{2}\). Wait, maybe I made a mistake. Wait, no: \(-2b + 2=-13\)
Subtract 2: \(-2b=-13 - 2=-15\)
Divide by -2: \(b=\frac{-15}{-2}=\frac{15}{2}\)? But the image shows \(b = -\frac{15}{2}\). Wait, maybe the original equation was \(|-2b - 2| = 13\)? No, the equation is \(|-2b + 2| = 13\). Wait, perhaps there's a typo in the image, but according to the standard solution:

Wait, let's do it again. For \(|-2b + 2| = 13\):

Case 1: \(-2b + 2 = 13\)
\(-2b=13 - 2 = 11\)
\(b = -\frac{11}{2}\)

Case 2: \(-2b + 2=-13\)
\(-2b=-13 - 2=-15\)
\(b=\frac{-15}{-2}=\frac{15}{2}\). But the image has \(b = -\frac{15}{2}\), which is incorrect. However, following the steps for the given equation \(|-2b + 2| = 13\), the correct solutions are \(b = -\frac{11}{2}\) and \(b=\frac{15}{2}\). But if we assume that there was a sign error in the equation (e.g., \(|-2b - 2| = 13\)), then:

Case 1: \(-2b - 2 = 13\)
\(-2b=15\)
\(b = -\frac{15}{2}\)

Case 2: \(-2b - 2=-13\)
\(-2b=-11\)
\(b=\frac{11}{2}\). But this is just to match the image. However, based on the equation \(|-2b + 2| = 13\) as written, the solutions are \(b = -\frac{11}{2}\) and \(b=\frac{15}{2}\). But since the image has \(b = -\frac{11}{2}\) and \(b = -\frac{15}{2}\), there might be a typo in the equation (maybe \(|-2b - 2| = 13\)). But following the steps for the equation \(|-2b + 2| = 13\):

The two solutions are \(b = -\frac{11}{2}\) and \(b=\frac{15}{2}\). But if we consider the image's answers, perhaps the equation was \(|-2b - 2| = 13\). Let's check that equation:

For \(|-2b - 2| = 13\):

Case 1: \(-2b - 2 = 13\)
\(-2b=15\)
\(b = -\frac{15}{2}\)

Case 2: \(-2b - 2=-13\)
\(-2b=-11\)
\(b=\frac{11}{2}\). No, that doesn't match. Alternatively, maybe the equation is \(|2b + 2| = 13\):

Case 1: \(2b + 2 = 13\)
\(2b=11\)
\(b=\frac{11}{2}\)

Case 2: \(2b + 2=-13\)
\(2b=-15\)
\(b = -\frac{15}{2}\). Ah! So maybe the original equation was \(|2b + 2| = 13\) (a sign error in the coefficient of \(b\)). If we take the equation as \(|2b + 2| = 13\), then:

Case 1: \(2b + 2 = 13\)
\(2b=11\)
\(b=\frac{11}{2}\) (no, not matching the image). Wait, the image has \(b = -\frac{11}{2}\) and \(b = -\frac{15}{2}\). Let's solve for \(b\) in the image's answers. Let's take \(b = -\frac{11}{2}\):

Plug into \(|-2b + 2|\): \(|-2\times(-\frac{11}{2})+2|=|11 + 2| = 13\), which works.

For \(b = -\frac{15}{2}\): \(|-2\times(-\frac{15}{2})+2|=|15 + 2| = 17 eq13\). Wait, that's not correct. Wait, no—wait, if the equation was \(|-2b - 2| = 13\):

For \(b = -\frac{11}{2}\): \(|-2\times(-\frac{11}{2})-2|=|11 - 2| = 9 eq13\).

Wait, I think there's a mistake in the image's second solution. But according to the correct method for \(|-2b + 2| = 13\), the solutions are \(b = -\frac{11}{2}\) and \(b=\frac{15}{2}\). However, if we follow the steps for the equation as given, the correct solutions are:

From \(|-2b + 2| = 13\):

1. \(-2b + 2 = 13\) gives \(b = -\frac{11}{2}\)
2. \(-2b + 2=-13\) gives \(b=\frac{15}{2}\)

Answer:

The correct solutions are \(b = -\frac{11}{2}\) and \(b=\frac{15}{2}\). (Note: The second solution in the image is incorrect. If we assume a typo in the equation, for example, if the equation was \(|-2b - 2| = 13\), the solutions would be different, but based on the given equation \(|-2b + 2| = 13\), the solutions are \(b = -\frac{11}{2}\) and \(b=\frac{15}{2}\).)