https://file.aiquickdraw.com/mathbot/file/1729051551588k9svlafm.jpg

The inequality given in the image is:

\[
|5x + \frac{1}{2} - 4| < 8
\]

We need to solve this absolute value inequality step by step.

### Step 1: Simplify the expression inside the absolute value.

\[
5x + \frac{1}{2} - 4
\]

Convert \( -4 \) into a fraction with denominator 2:

\[
5x + \frac{1}{2} - \frac{8}{2} = 5x - \frac{7}{2}
\]

So, the inequality becomes:

\[
|5x - \frac{7}{2}| < 8
\]

### Step 2: Remove the absolute value by writing two inequalities.

When we solve absolute value inequalities, we have two cases:

1. \( 5x - \frac{7}{2} < 8 \)
2. \( -(5x - \frac{7}{2}) < 8 \) (or equivalently \( 5x - \frac{7}{2} > -8 \))

### Step 3: Solve each inequality separately.

#### Case 1: Solve \( 5x - \frac{7}{2} < 8 \)

Add \( \frac{7}{2} \) to both sides:

\[
5x < 8 + \frac{7}{2}
\]

Convert 8 into a fraction:

\[
5x < \frac{16}{2} + \frac{7}{2} = \frac{23}{2}
\]

Now, divide both sides by 5:

\[
x < \frac{23}{10}
\]

#### Case 2: Solve \( 5x - \frac{7}{2} > -8 \)

Add \( \frac{7}{2} \) to both sides:

\[
5x > -8 + \frac{7}{2}
\]

Convert \( -8 \) into a fraction:

\[
5x > -\frac{16}{2} + \frac{7}{2} = -\frac{9}{2}
\]

Now, divide both sides by 5:

\[
x > -\frac{9}{10}
\]

### Step 4: Combine the results.

The solution to the inequality is:

\[
-\frac{9}{10} < x < \frac{23}{10}
\]

### Final Answer:

The solution set is:

\[
(-\frac{9}{10}, \frac{23}{10})
\]

Thus, the correct choice is A, and the solution set in interval notation is \( \left( -\frac{9}{10}, \frac{23}{10} \right) \).

---

Would you like more details or have any questions?

Here are 5 follow-up questions:
1. How do you handle absolute value inequalities that involve "greater than" instead of "less than"?
2. What happens if you add or subtract a constant inside the absolute value?
3. How can you represent the solution to inequalities using a number line?
4. What is the difference between strict inequalities (like "<") and non-strict ones (like "≤")?
5. How does multiplying or dividing by a negative number affect inequalities?

**Tip**: When solving inequalities, be careful with signs, especially when multiplying or dividing by negative numbers, as this reverses the inequality.

Solve the inequality |5x + 1/2 - 4| < 8 and determine the solution set.

Learn how to solve the absolute value inequality |5x + 1/2 - 4| < 8 with a step-by-step solution. This problem covers algebraic concepts involving absolute value inequalities and includes interval notation for the solution set. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation