Question

solve for x: 2|11x + 1| + 1 = 4
a) ( x = \frac{1}{22}, x = -\frac{5}{22} )
b) ( x = \frac{1}{22} )
c) ( x = -\frac{5}{22} )
d) ( x = \frac{1}{11}, x = -\frac{5}{22} )
e) ( x = \frac{1}{11} )
f) none of the above.

Explanation:

Step1: Isolate the absolute value

Subtract 1 from both sides of the equation \(2|11x + 1|+1 = 4\) to get \(2|11x + 1|=4 - 1\).
Simplify the right side: \(2|11x + 1| = 3\).
Then divide both sides by 2: \(|11x + 1|=\frac{3}{2}\).

Step2: Solve the absolute value equation

The absolute value equation \(|11x + 1|=\frac{3}{2}\) gives two cases:

Case 1: \(11x+1=\frac{3}{2}\)

Subtract 1 from both sides: \(11x=\frac{3}{2}-1\).
Simplify the right side: \(11x=\frac{3 - 2}{2}=\frac{1}{2}\).
Divide both sides by 11: \(x=\frac{1}{2\times11}=\frac{1}{22}\).

Case 2: \(11x + 1=-\frac{3}{2}\)

Subtract 1 from both sides: \(11x=-\frac{3}{2}-1\).
Simplify the right side: \(11x=-\frac{3 + 2}{2}=-\frac{5}{2}\).
Divide both sides by 11: \(x=-\frac{5}{2\times11}=-\frac{5}{22}\).

So the solutions are \(x = \frac{1}{22}\) and \(x=-\frac{5}{22}\), which matches option d (since option d is \(x=\frac{1}{11}\) (wait, no, let's re - check the options. Wait the user's option d: "x = 1/11: x = -5/22" (maybe a typo, but from our solution, the solutions are \(x=\frac{1}{22}\) and \(x = -\frac{5}{22}\)? Wait no, wait in step 2 case 1:

Wait \(11x+1=\frac{3}{2}\)

\(11x=\frac{3}{2}-1=\frac{3 - 2}{2}=\frac{1}{2}\)

\(x=\frac{1}{2\times11}=\frac{1}{22}\)

Case 2: \(11x + 1=-\frac{3}{2}\)

\(11x=-\frac{3}{2}-1=-\frac{3 + 2}{2}=-\frac{5}{2}\)

\(x=-\frac{5}{2\times11}=-\frac{5}{22}\)

Looking at the options:

Option d: "x = 1/11: x = -5/22" (maybe a typo, perhaps it's \(x=\frac{1}{22}\) and \(x = -\frac{5}{22}\), but among the given options, option d has \(x = -\frac{5}{22}\) and a wrong first solution? Wait no, maybe I made a mistake in the original equation. Wait the original equation: "Solve for x: 2|11x + 1|+1 = 4"

Wait let's re - do the first step:

\(2|11x + 1|+1 = 4\)

Subtract 1: \(2|11x + 1|=3\)

Divide by 2: \(|11x + 1|=\frac{3}{2}\)

Then \(11x+1=\frac{3}{2}\) or \(11x + 1=-\frac{3}{2}\)

First equation: \(11x=\frac{3}{2}-1=\frac{1}{2}\), \(x=\frac{1}{22}\)

Second equation: \(11x=-\frac{3}{2}-1=-\frac{5}{2}\), \(x = -\frac{5}{22}\)

Now looking at the options:

a) \(x=\frac{1}{22},x = -\frac{5}{22}\) – wait no, option a is "x = 1/22, x = -5/22"? Wait the user's option a: "x = 1/22, x = -5/22"? Wait the user's image:

a) \(x=\frac{1}{22},x = -\frac{5}{22}\)

b) \(x=\frac{1}{22}\)

c) \(x=-\frac{5}{22}\)

d) \(x=\frac{1}{11}:x = -\frac{5}{22}\)

e) \(x=\frac{1}{11}\)

f) None of the above.

Wait maybe the original problem's option a is correct? Wait no, in the user's image, option a: "x = 1/22, x = -5/22" (the user wrote "x = 1/22, x = -5/22" for option a? Wait the user's text:

"a) x = 1/22, x = -5/22"

"b) x = 1/22"

"c) x = -5/22"

"d) x = 1/11: x = -5/22"

"e) x = 1/11"

"f) None of the above"

Ah, I see, so option a is \(x=\frac{1}{22}\) and \(x = -\frac{5}{22}\), which are the two solutions we found. But in the image, the selected option is b, but that's incorrect. Wait maybe I misread the equation. Wait the original equation: "2|11x + 1|+1 = 4"? Or is it "2|11x + 1|+11 = 4"? No, the user wrote "2|11x + 1|+1 = 4".

Wait let's recalculate:

\(2|11x + 1|+1 = 4\)

\(2|11x + 1|=3\)

\(|11x + 1|=\frac{3}{2}\)

\(11x+1=\frac{3}{2}\) or \(11x + 1=-\frac{3}{2}\)

First solution:

\(11x=\frac{3}{2}-1=\frac{1}{2}\)

\(x=\frac{1}{22}\)

Second solution:

\(11x=-\frac{3}{2}-1=-\frac{5}{2}\)

\(x = -\frac{5}{22}\)

So the correct solutions are \(x=\frac{1}{22}\) and \(x = -\frac{5}{22}\), which is option a. But in the image, option b is selected, which is wrong.

Wait maybe the original equation is \(2|11x + 1|+11 = 4\)? Let's check:

\(2|11x + 1|+11 = 4\)

\(2|11x + 1|=4 - 11=-7\), which is impossible since absolute value is non - negative, so \(2|11x + 1|\geq0\), so \(2|11x + 1|+11\geq11>4\), no solution. But that's not the case.

Wait the user's image: "Solve for x: 2|11x + 1|+1 = 4"

So the correct solutions are \(x=\frac{1}{22}\) and \(x = -\frac{5}{22}\), which is option a. But in the user's image, option a is "x = 1/22, x = -5/22", so the correct answer is option a.

Answer:

A. \(x=\frac{1}{22},x = -\frac{5}{22}\)