Math Problem Statement

Solve
| 2 x + 3 | + 4 < 5 and write the solution in interval notation.

(If there is no solution, enter your answer as
∅ .)

Provide your answer below:

Solution

To solve the inequality:

2x+3+4<5|2x + 3| + 4 < 5

Step 1: Subtract 4 from both sides of the inequality:

2x+3<54|2x + 3| < 5 - 4

2x+3<1|2x + 3| < 1

Step 2: Now, solve the absolute value inequality. This implies that:

1<2x+3<1-1 < 2x + 3 < 1

Step 3: Solve for xx by subtracting 3 from all parts of the inequality:

13<2x<13-1 - 3 < 2x < 1 - 3

4<2x<2-4 < 2x < -2

Step 4: Divide the entire inequality by 2:

2<x<1-2 < x < -1

Thus, the solution in interval notation is:

(2,1)(-2, -1)


Would you like further details or have any questions? Here are 5 related questions:

  1. What does the absolute value represent in an inequality?
  2. How would you solve an absolute value inequality with a greater-than symbol?
  3. What happens when you have multiple absolute value terms in an inequality?
  4. How would you graph the solution of an absolute value inequality?
  5. How do you handle absolute values in equations versus inequalities?

Tip: Remember that for absolute value inequalities, you always break them down into two cases—one for the positive and one for the negative scenario.

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Math Problem Analysis

Mathematical Concepts

Inequalities
Absolute Value Inequalities

Formulas

|A| < B
Solving linear inequalities

Theorems

Absolute value inequality theorem

Suitable Grade Level

Grades 8-10