Question

1. $-2|x - 5| + 4 = 1$

Explanation:

Step1: Isolate absolute value term

Subtract 4 from both sides.
$-2|x-5| = 1 - 4$
$-2|x-5| = -3$

Step2: Solve for absolute value

Divide both sides by -2.
$|x-5| = \frac{-3}{-2}$
$|x-5| = \frac{3}{2}$

Step3: Split into two equations

Absolute value gives two cases.
$x-5 = \frac{3}{2}$ or $x-5 = -\frac{3}{2}$

Step4: Solve first equation

Add 5 to both sides.
$x = \frac{3}{2} + 5 = \frac{3}{2} + \frac{10}{2} = \frac{13}{2}$

Step5: Solve second equation

Add 5 to both sides.
$x = -\frac{3}{2} + 5 = -\frac{3}{2} + \frac{10}{2} = \frac{7}{2}$

Answer:

$x=\frac{13}{2}$ or $x=\frac{7}{2}$