Question

solve the following absolute value inequality. graph the solution set on a real number line.

\\5|x - 4| + 1 < 6\\

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution set is \\(\box\\).
(simplify your answer. type your answer in interval notation. use integers or fractions for any numbers in the expression.)
b. the solution set is \\(\\{\\}\\) or \\(\varnothing\\).

choose the correct graph below.

a. a number line showing an open interval from 3 to 5 with parentheses at 3 and 5.
c. a number line showing two rays extending outwards from 3 and 5 with parentheses at 3 and 5.
e. a number line shaded completely.

Explanation:

🆕 New Concept Discovered: Solving Absolute Value Inequalities
Finding the range of values within a certain distance

Step 1: Isolate the absolute value expression

Subtract \( 1 \) from both sides:

$$ 5|x - 4| < 5 $$

Divide both sides by \( 5 \):

$$ |x - 4| < 1 $$

Step 2: Rewrite as a compound inequality

An absolute value inequality of the form \( |u| < a \) (where \( a > 0 \)) means the distance from \( 0 \) is less than \( a \). This translates to:

$$ -1 < x - 4 < 1 $$

Step 3: Solve for \( x \)

Add \( 4 \) to all parts of the inequality:

$$ -1 + 4 < x < 1 + 4 $$

$$ 3 < x < 5 $$

In interval notation, this is written as:

$$ (3, 5) $$

Step 4: Identify the correct graph

The solution set \( (3, 5) \) represents all numbers strictly between \( 3 \) and \( 5 \). On a number line, this is represented by parentheses at \( 3 \) and \( 5 \) with shading in between them. This matches graph A.

Answer:

The correct choice for the solution set is A.
The solution set is \( (3, 5) \).

The correct graph is A.