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Magazine Chapter 0: Problem 54
Solve each rational inequality and express the solution set in interval notation. $$\frac{-2 t-t^{2}}{4-t} \geq t$$
Short Answer
Expert verified
The solution set is \((-
finity, 0] \cup (4,
finity)\).
Step by step solution
01
Move All Terms to One Side
Begin by writing the inequality with all terms on one side: \( \frac{-2t - t^2}{4-t} - t \geq 0 \). This sets up the inequality for simplification.
02
Find a Common Denominator
For the expression \( \frac{-2t - t^2}{4-t} - t \), find a common denominator, which is \( 4-t \). Rewrite the inequality as: \( \frac{-2t - t^2 - t(4-t)}{4-t} \geq 0 \).
03
Simplify the Numerator
Distribute and simplify the numerator: \( -2t - t^2 - 4t + t^2 = -6t \). This makes the inequality: \( \frac{-6t}{4-t} \geq 0 \).
04
Find Critical Points
To find the critical points, set the numerator and denominator equal to zero. For the numerator \(-6t = 0\), the solution is \(t=0\). For the denominator \(4-t = 0\), the solution is \(t=4\). These critical points divide the number line into intervals.
05
Analyze the Intervals
Test each interval: \((-\infty, 0)\), \((0, 4)\), and \((4, \infty)\) to determine where the inequality holds.1. For \( t < 0 \), e.g., \( t = -1 \), the expression is positive.2. For \( 0 < t < 4 \), e.g., \( t = 2 \), the expression is negative.3. For \( t > 4 \), e.g., \( t = 5 \), the expression is positive.
06
Determine Undefined Points and Boundaries
At \( t = 0 \), the expression is zero and holds. At \( t = 4 \), the expression is undefined due to division by zero. Thus, \( t = 4 \) cannot be included in the solution set.
07
Write the Solution in Interval Notation
Combine the intervals where the inequality holds: \((-finity, 0] \cup (4, finity)\). This represents the values of \( t \) for which the inequality holds.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a concise way of expressing a set of solutions to inequalities. It helps us understand which values satisfy the inequality. For example, the interval \((-\infty, 0]\) represents all numbers less than or equal to zero, while \((4, \infty)\) represents all numbers greater than four.
Each part of the interval notation has a specific meaning:
The union symbol \(\cup\) is used to combine different intervals where the inequality holds true. It acts like a mathematical "OR," showing that solutions can come from either interval or both.
Each part of the interval notation has a specific meaning:
- Parentheses \(()\) indicate that the endpoint is not included in the interval, making it an 'open' interval at that side.
- Brackets \([]\) indicate that the endpoint is included, marking it as a 'closed' interval on that side.
The union symbol \(\cup\) is used to combine different intervals where the inequality holds true. It acts like a mathematical "OR," showing that solutions can come from either interval or both.
Critical Points
Critical points are important values obtained while solving inequalities, which help determine where the sign of the function changes. Identifying these points is crucial because they divide the number line into sections where the inequality may hold true.
To find critical points, first set the numerator and denominator of the rational expression equal to zero separately. In this exercise, solving
These two critical points divide the number line into intervals. We must then test each interval to see if the inequality holds true in those sections. Critical points often indicate where to expect changes in the inequality's direction. Understanding this helps in identifying the correct solution set.
To find critical points, first set the numerator and denominator of the rational expression equal to zero separately. In this exercise, solving
- \(-6t = 0\) gives us the critical point \(t=0\).
- \(4-t = 0\) gives \(t=4\) as another critical point.
These two critical points divide the number line into intervals. We must then test each interval to see if the inequality holds true in those sections. Critical points often indicate where to expect changes in the inequality's direction. Understanding this helps in identifying the correct solution set.
Common Denominator
When dealing with rational expressions, finding a common denominator is essential to simplify expressions and solve inequalities effectively. The common denominator allows us to combine terms into one fraction, which is easier to manage.
For example, in this problem, the inequality \(\frac{-2t - t^2}{4-t} - t \geq 0\) requires finding a common denominator to simplify the expression. The denominator here is \(4-t\).
This step involves:
Once the terms are expressed with a common denominator, it becomes straightforward to simplify the problem and find the inequality's solution.
For example, in this problem, the inequality \(\frac{-2t - t^2}{4-t} - t \geq 0\) requires finding a common denominator to simplify the expression. The denominator here is \(4-t\).
This step involves:
- Rewriting the whole inequality so that all terms share the same denominator.
- Combining terms simplifies expression manipulation and reveals the inequality's core structure.
Once the terms are expressed with a common denominator, it becomes straightforward to simplify the problem and find the inequality's solution.
