To solve the equation
[tex]\[
\left|\frac{7z - 5}{5z + 2}\right| = 1,
\][/tex]
we will break it down into two separate equations because the absolute value equation [tex]\( |A| = 1 \)[/tex] implies that either [tex]\( A = 1 \)[/tex] or [tex]\( A = -1 \)[/tex].
### Case 1: [tex]\(\frac{7z - 5}{5z + 2} = 1\)[/tex]
1. Set up the equation:
[tex]\[
\frac{7z - 5}{5z + 2} = 1
\][/tex]
2. Clear the fraction by multiplying both sides by [tex]\( 5z + 2 \)[/tex]:
[tex]\[
7z - 5 = 5z + 2
\][/tex]
3. Move all terms involving [tex]\( z \)[/tex] to one side and constants to the other:
[tex]\[
7z - 5z = 2 + 5
\][/tex]
[tex]\[
2z = 7
\][/tex]
4. Solve for [tex]\( z \)[/tex]:
[tex]\[
z = \frac{7}{2}
\][/tex]
### Case 2: [tex]\(\frac{7z - 5}{5z + 2} = -1\)[/tex]
1. Set up the equation:
[tex]\[
\frac{7z - 5}{5z + 2} = -1
\][/tex]
2. Clear the fraction by multiplying both sides by [tex]\( 5z + 2 \)[/tex]:
[tex]\[
7z - 5 = - (5z + 2)
\][/tex]
3. Distribute the negative sign on the right:
[tex]\[
7z - 5 = -5z - 2
\][/tex]
4. Move all terms involving [tex]\( z \)[/tex] to one side and constants to the other:
[tex]\[
7z + 5z = -2 + 5
\][/tex]
[tex]\[
12z = 3
\][/tex]
5. Solve for [tex]\( z \)[/tex]:
[tex]\[
z = \frac{3}{12} = \frac{1}{4}
\][/tex]
### Solutions:
The solutions to the original equation are:
[tex]\[
z = \frac{7}{2}, \frac{1}{4}
\][/tex]
Thus, the final answer is:
[tex]\[
z = \frac{7}{2}, \frac{1}{4}
\][/tex]
