a. To find: The logical first step in solving equation |2x-1|=5 b.

a) Approach:
Given below are the steps to solve an absolute value equation:
1. If fractions are present, multiply both sides of the equation by the LCD of the fractions.
2. Use the addition property of equality to rewrite the equation so that the abolute value is on one side of the equation and constant terms are on the other side.
3. Use the following property of absolute value:
${\displaystyle \left|x\right|=c}$ is equivalent to ${\displaystyle \left|x\right|=\pm c}$
This property says that to solve an absolute value equation, we must solve two separate equations.
For example, ${\displaystyle \left|x\right|=6}$ is equivalent to two equations ${\displaystyle x=6\text{and}x=-6}$
Calculation:
The given equation is ${\displaystyle |2x-1|=5}$.
In this equation, only absolute value term is one side of the equation.
Hence, we can use the property of absolute value to obtain two equations.
Thus, ${\displaystyle |2x-1|=5}$ is equivalent to ${\displaystyle 2x-1=5\text{and}2x-1=-5}$
Final solution:
The logical first step in solving the equation ${\displaystyle |2x-1|=5}$ will be use the property of absolute value which says ${\displaystyle \left|x\right|=c}$ is equivalent to ${\displaystyle \left|x\right|=\pm c}$.
Hence, the given equation ${\displaystyle |2x-1|=5}$ is equivalent to ${\displaystyle 2x-1=5\text{and}2x-1=-5}$
b) Approach:
Absolute value inequality:
The table given below shows various inequalities and their solution sets:
Inequality Solution Set:
${\displaystyle \left|x\right|<c-c<x<c}$
${\displaystyle \left|x\right|\le c-c\le x\le c}$
${\displaystyle \left|x\right|>cx<-c\text{or}x>c}$
${\displaystyle \left|x\right|\ge cx\le -c\text{or}x\ge c}$
The above properties hold true even if x is replaced by an algebraic expression.
Calculation:
The given inequality is ${\displaystyle |3x+2|\le 8}$
As it is clear from the above table, solution set of inequality can be written as:
${\displaystyle -8\le 3x+2\le 8}$
Once we have written the inequality in this form, we can split write it in terms of two inequalities:
${\displaystyle 3x+2\ge -8\text{and}3x+2\le 8}$
Final solution:
The first logical step in solving the inequality ${\displaystyle |3x+2|\le 8}$ would be to use the properties of absolute value inequalities and obtain the following inequalities:
${\displaystyle 3x+2\ge -8\text{and}3x+2\le 8}$