Math Problem Statement

solve symbolically 4x+30\frac{4}{x+3}\ge0

Solution

To solve the inequality

4x+30,\frac{4}{x+3} \geq 0,

follow these steps:

Step 1: Understand the inequality

The inequality will hold when the numerator is non-negative, and the denominator does not cause division by zero.

  1. The numerator 44 is always positive.
  2. The denominator x+3x + 3 will determine the sign of the fraction.

Step 2: Analyze the denominator

The denominator, x+3x + 3, affects the sign of the fraction:

  • When x+3>0x + 3 > 0, the fraction is positive.
  • When x+3=0x + 3 = 0, the fraction is undefined.
  • When x+3<0x + 3 < 0, the fraction is negative.

Step 3: Solve x+3>0x + 3 > 0

x+3>0    x>3.x + 3 > 0 \implies x > -3.

Step 4: Write the solution

  • For x>3x > -3, the fraction is positive, so 4x+30\frac{4}{x+3} \geq 0 is true.
  • For x=3x = -3, the fraction is undefined.
  • For x<3x < -3, the fraction is negative, so the inequality is false.

Final Solution:

x(3,).x \in (-3, \infty).

This is the solution in interval notation.


Would you like a detailed explanation of any step?

Here are 5 follow-up questions:

  1. How does the inequality change if 4x+30\frac{-4}{x+3} \geq 0?
  2. Can we include x=3x = -3 in the solution?
  3. What happens if the numerator also depends on xx?
  4. How would this inequality change if the fraction were 0\leq 0 instead?
  5. What are some applications of such inequalities in real-life scenarios?

Tip: Always check for division by zero when solving rational inequalities.

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Math Problem Analysis

Mathematical Concepts

Inequalities
Rational Expressions

Formulas

Sign analysis of rational expressions
Interval notation

Theorems

Properties of rational functions
Sign chart method for inequalities

Suitable Grade Level

Grades 10-12