Math Problem Statement

Solve the following inequality: 12 / (x + 3) ≤ 4. Write your answer using interval notation.

Solution

The inequality given is:

12x+34\frac{12}{x + 3} \leq 4

Step-by-Step Solution:

  1. Isolate the fraction: We start by eliminating the fraction. Multiply both sides of the inequality by x+3x + 3, but be cautious: if x+3x + 3 is negative, we need to reverse the inequality sign.

    First, assume x+3>0x + 3 > 0 (so no sign change). Then:

    124(x+3)12 \leq 4(x + 3)

  2. Simplify the inequality: Expand the right-hand side:

    124x+1212 \leq 4x + 12

  3. Subtract 12 from both sides:

    04x0 \leq 4x

  4. Solve for xx:

    0x0 \leq x

    Or equivalently:

    x0x \geq 0

  5. Consider the case where x+3<0x + 3 < 0: If x+3x + 3 is negative, multiplying by it would flip the inequality sign. The inequality would become:

    124(x+3)12 \geq 4(x + 3)

    Simplifying this:

    124x+1212 \geq 4x + 12

    Subtracting 12 from both sides:

    04x0 \geq 4x

    Dividing by 4:

    0x0 \geq x

    Or:

    x0x \leq 0

  6. Combine both cases: From both cases, we get that x0x \geq 0 from the first case and x0x \leq 0 from the second. Therefore, the solution is:

    x=0x = 0

Final Answer in Interval Notation:

{0}\{ 0 \}

Let me know if you need any further clarification or a detailed explanation of any of the steps!


Would you like more details or have any questions? Here are some related questions:

  1. What if the inequality was strict (<<) instead of \leq?
  2. How do we handle inequalities with absolute values?
  3. What happens if the inequality involves a quadratic expression?
  4. Can this method be applied to systems of inequalities?
  5. How would the solution change if we had 12x+34\frac{12}{x+3} \geq 4?

Tip: Always check for cases when multiplying or dividing inequalities by variables, as the inequality sign may change depending on whether the variable is positive or negative.

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Math Problem Analysis

Mathematical Concepts

Inequalities
Rational Expressions

Formulas

Inequality rules: When multiplying/dividing by a negative, flip the inequality
Solving rational expressions: Cross-multiply considering positive and negative cases

Theorems

Multiplication rule for inequalities
Case analysis when solving inequalities involving rational expressions

Suitable Grade Level

Grades 9-11