Math Problem Statement

|2x - 1| < 4

Solution

The image contains the inequality:

2x1<4|2x - 1| < 4

This is an absolute value inequality, which can be solved by considering the two cases for the absolute value expression:

  1. 2x1<42x - 1 < 4
  2. 2x1>42x - 1 > -4

Let's solve these step by step.

Case 1: 2x1<42x - 1 < 4

Add 1 to both sides:

2x<52x < 5

Now divide by 2:

x<52orx<2.5x < \frac{5}{2} \quad \text{or} \quad x < 2.5

Case 2: 2x1>42x - 1 > -4

Add 1 to both sides:

2x>32x > -3

Now divide by 2:

x>32orx>1.5x > -\frac{3}{2} \quad \text{or} \quad x > -1.5

Final Solution:

Combining the results of both cases, the solution to the inequality is:

32<x<52-\frac{3}{2} < x < \frac{5}{2}

Thus, the solution set is x(32,52)x \in \left(-\frac{3}{2}, \frac{5}{2}\right).

Would you like further details or clarifications on any part of the solution?

Here are 5 related questions to deepen your understanding:

  1. How do you solve absolute value inequalities that involve a greater-than sign (e.g., x3>5|x - 3| > 5)?
  2. How would you solve an absolute value inequality if it had both variables inside the absolute value?
  3. What is the geometric meaning of absolute value inequalities on a number line?
  4. Can you convert the solution of an absolute value inequality into interval notation?
  5. How would the inequality change if the constant on the right side was negative?

Tip: When solving absolute value inequalities, always break it down into two cases—one for the positive expression and one for the negative expression.

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Math Problem Analysis

Mathematical Concepts

Absolute Value Inequalities
Linear Inequalities

Formulas

|a| < b means -b < a < b

Theorems

Properties of Absolute Values

Suitable Grade Level

Grades 8-10