https://file.aiquickdraw.com/mathbot/file/1727088869190hj87ghrx.jpg

The image contains several absolute value inequalities. Let's break down the steps for solving these types of inequalities, and then I'll solve a few of them as examples.

### General approach to solving absolute value inequalities:

For an absolute value inequality \( |A| \geq B \):
- It splits into two inequalities: 
  1. \( A \geq B \)
  2. \( A \leq -B \)

For an absolute value inequality \( |A| \leq B \):
- It becomes a compound inequality: 
  \( -B \leq A \leq B \)

Now I'll solve a few examples from the image:

---

### Example 1: Problem 35
\[ |x - 2| \geq 5 \]

This splits into two inequalities:
1. \( x - 2 \geq 5 \) ⟶ \( x \geq 7 \)
2. \( x - 2 \leq -5 \) ⟶ \( x \leq -3 \)

**Solution:** \( x \leq -3 \) or \( x \geq 7 \)

---

### Example 2: Problem 36
\[ |x + 2| < 1 \]

This becomes a compound inequality:
\[ -1 < x + 2 < 1 \]

Now solve for \( x \):
1. Subtract 2 from all parts:
   \[ -3 < x < -1 \]

**Solution:** \( -3 < x < -1 \)

---

### Example 3: Problem 37
\[ |4x + 5| \leq 10 \]

This becomes a compound inequality:
\[ -10 \leq 4x + 5 \leq 10 \]

Now solve for \( x \):
1. Subtract 5 from all parts:
   \[ -15 \leq 4x \leq 5 \]
2. Divide by 4:
   \[ -\frac{15}{4} \leq x \leq \frac{5}{4} \]

**Solution:** \( -\frac{15}{4} \leq x \leq \frac{5}{4} \)

---

### Example 4: Problem 38
\[ |2x - 1| > 2 \]

This splits into two inequalities:
1. \( 2x - 1 > 2 \) ⟶ \( 2x > 3 \) ⟶ \( x > \frac{3}{2} \)
2. \( 2x - 1 < -2 \) ⟶ \( 2x < -1 \) ⟶ \( x < -\frac{1}{2} \)

**Solution:** \( x > \frac{3}{2} \) or \( x < -\frac{1}{2} \)

---

Would you like to see the steps for the remaining problems, or ask about specific ones? 

### Related questions:
1. How do we solve absolute value inequalities involving fractions (like problem 39)?
2. What happens if the absolute value inequality has a strict inequality (e.g., \( > \) or \( < \))?
3. How does dividing or multiplying by negative numbers affect inequalities?
4. How can you check your solution for absolute value inequalities graphically?
5. How do absolute value equations differ from inequalities?

### Tip:
Always remember to split absolute value inequalities correctly into their two parts when solving, and apply the correct inequality sign when breaking them down.

Solve the absolute value inequalities from the provided set of problems.

Learn how to solve absolute value inequalities with step-by-step examples. Problems involve algebraic manipulation and inequality rules. 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