Problem 52 - Algebra A Combined Function

Question

Solve each inequality. Write the solution set in interval notation. $$ \frac{p}{p+4} \leq 3 p $$

Step-by-Step Solution

Answer

\((-\frac{11}{3}, 0]\)

Step 1: Subtract One Side from the Other

To solve the inequality \( \frac{p}{p+4} \leq 3p \), we first move all terms to one side, leading to a single inequality with zero on the other side:\[ \frac{p}{p+4} - 3p \leq 0. \]

Step 2: Simplify the Expression

Combine the terms over a common denominator:\[ \frac{p - 3p(p+4)}{p+4} \leq 0. \] Simplifying the numerator, we get:\[ p - 3p^2 - 12p = -3p^2 - 11p. \] Thus, the inequality becomes:\[ \frac{-3p^2 - 11p}{p+4} \leq 0. \]

Step 3: Factorize the Numerator

Factor the quadratic expression in the numerator:\[ -3p^2 - 11p = -p(3p + 11). \] Substitute back into the inequality:\[ \frac{-p(3p + 11)}{p+4} \leq 0. \] This expresses the inequality in terms of its critical points.

Step 4: Find Critical Points

Identify the critical points by setting each factor equal to zero:1. \( -p = 0 \) gives \( p = 0 \).2. \( 3p + 11 = 0 \) gives \( p = -\frac{11}{3} \).3. The denominator is zero when \( p + 4 = 0 \), giving \( p = -4 \).Thus, the critical points are \( p = 0 \), \( p = -\frac{11}{3} \), and \( p = -4 \).

Step 5: Analyze Intervals Between Critical Points

The critical points divide the number line into intervals: \((-\infty, -4)\), \((-4, -\frac{11}{3})\), \((-\frac{11}{3}, 0)\), and \((0, \infty)\). Test each interval in the inequality:1. In \((-\infty, -4)\), choose \(p = -5\).2. In \((-4, -\frac{11}{3})\), choose \(p = -3.5\).3. In \((-\frac{11}{3}, 0)\), choose \(p = -1\).4. In \((0, \infty)\), choose \(p = 1\).Calculate for each interval to determine where the inequality holds, ensuring to check each interval separately with the sign analysis of the expression.

Step 6: Conclusion and Interval Notation

After analyzing, the inequality holds for the intervals where the expression is \( \leq \) 0. These are \([-\frac{11}{3}, 0]\) excluding the point \(p = -4\) where the expression is undefined. Thus, in interval notation, the solution set is:\[ \left(-\frac{11}{3}, 0\right]. \]

Key Concepts

Quadratic Factorization

Quadratic factorization is a vital step in solving many algebraic inequalities, especially those involving rational expressions. In our exercise, we encountered the quadratic expression \(-3p^2 - 11p\).
In order to factorize it, we look for common factors in the terms of the quadratic expression. Here, we can factor out \(-p\), which simplifies the expression to: -\(-p(3p + 11)\).
Quadratic factorization is useful because it breaks down a polynomial into simpler components, making it easier to analyze and solve. Factorizing the expression helps us identify the critical points, essential for solving the inequality.

Interval Notation

Interval notation is a concise way to represent a set of numbers. It is especially useful when dealing with solutions to inequalities.
In this exercise, after determining where the inequality \(\frac{-p(3p + 11)}{p + 4} \leq 0\) holds, we express these ranges in interval notation.
For example, the solution \([-\frac{11}{3}, 0)\) states that \(p\) can take any value from \(-\frac{11}{3}\) to \(0\), but not including \(-4\), where the expression is undefined.
This notation makes it straightforward to understand which intervals satisfy the inequality.

Critical Points Analysis

Critical points analysis is performed by finding values that make each factor of a rational expression equal to zero or undefined.
In our inequality, critical points are determined by:

• Setting \(-p = 0\), finding \(p = 0\)
• Setting \(3p + 11 = 0\), finding \(p = -\frac{11}{3}\)
• Setting \(p + 4 = 0\), which makes the expression undefined at \(p = -4\)

The critical points \(0\), \(-\frac{11}{3}\), and \(-4\) divide the number line into intervals. Analyzing these intervals helps determine where the inequality holds true.

Rational Inequalities

Rational inequalities involve expressions where a polynomial is divided by another polynomial. Solving these inequalities often involves several steps:

• Combine terms to achieve a common denominator.
• Move terms to one side to simplify the inequality.
• Factorize and find critical points which divide the number line into intervals.
• Test each interval to check where inequality holds true, excluding points where the expression is undefined.

Our exercise involved the rational inequality \(\frac{-p(3p + 11)}{p + 4} \leq 0\). Solving this inequality required several techniques like simplifying, factorizing, and interval testing.
This systematic approach ensures that we consider all possible solutions and accurately depict them using interval notation.