Question

solve the following absolute value inequality. graph the solution set on

\\|3 - 2x| + 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\\).

Explanation:

🆕 New Concept Discovered: Solving Absolute Value Inequalities
Distance from zero is bounded by a value

Step 1: Isolate the absolute value expression

To solve an absolute value inequality, we must first get the absolute value term by itself on one side of the inequality.

Subtract \(1\) from both sides of the inequality:

$$ |3 - 2x| + 1 < 6 $$

$$ |3 - 2x| < 5 $$

Step 2: Rewrite as a compound inequality

An inequality of the form \(|u| < a\) (where \(a > 0\)) means the expression inside the absolute value, \(u\), is less than \(a\) units away from zero. This translates to a compound inequality:

$$ -a < u < a $$

Applying this to our inequality:

$$ -5 < 3 - 2x < 5 $$

Step 3: Solve the compound inequality

Now, we isolate \(x\) in the middle of the compound inequality.

First, subtract \(3\) from all three parts:

$$ -5 - 3 < -2x < 5 - 3 $$

$$ -8 < -2x < 2 $$

Next, divide all parts by \(-2\). Remember that dividing or multiplying an inequality by a negative number reverses the direction of the inequality signs:

$$ \frac{-8}{-2} > x > \frac{2}{-2} $$

$$ 4 > x > -1 $$

We can rewrite this in standard order (from least to greatest):

$$ -1 < x < 4 $$

Step 4: Write the solution in interval notation

The inequality \(-1 < x < 4\) represents all real numbers strictly between \(-1\) and \(4\). Since the inequalities are strict (using \(<\) rather than \(\le\)), we use parentheses:

$$ (-1, 4) $$

Answer:

A. The solution set is \((-1, 4)\).