Question

1. \\(|2x - 1| = 23\\)

1. \\(|x + 4| = 7\\)

1. \\(|2x + 5| + 8 = 23\\)

Explanation:

Problem 2: $|2x - 1| = 23$

Step1: Split absolute value cases

$2x - 1 = 23$ or $2x - 1 = -23$

Step2: Solve first equation

$2x = 23 + 1 = 24$
$x = \frac{24}{2} = 12$

Step3: Solve second equation

$2x = -23 + 1 = -22$
$x = \frac{-22}{2} = -11$

Problem 4: $|x + 4| = 7$

Step1: Split absolute value cases

$x + 4 = 7$ or $x + 4 = -7$

Step2: Solve first equation

$x = 7 - 4 = 3$

Step3: Solve second equation

$x = -7 - 4 = -11$

Problem 6: $|2x + 5| + 8 = 23$

Step1: Isolate absolute value term

$|2x + 5| = 23 - 8 = 15$

Step2: Split absolute value cases

$2x + 5 = 15$ or $2x + 5 = -15$

Step3: Solve first equation

$2x = 15 - 5 = 10$
$x = \frac{10}{2} = 5$

Step4: Solve second equation

$2x = -15 - 5 = -20$
$x = \frac{-20}{2} = -10$

Answer:

Problem 2: $x = 12$ or $x = -11$
Problem 4: $x = 3$ or $x = -11$
Problem 6: $x = 5$ or $x = -10$