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Chapter 7: Problem 376

Create a rational inequality whose solution is \((-\infty,-2] \cup[4, \infty)\).

Short Answer

Expert verified
\( \frac{1}{(x+2)(x-4)} \leq 0 \)

Step by step solution

01

Analyze the given solution set

The solution set is \((-\rightarrow, -2] \cup [4, \rightarrow)\). This implies that the inequality has critical points at -2 and 4 where the inequality changes behavior.
02

Construct a rational expression

For the solution set to be involved in a rational inequality, its critical points must create intervals that either include the points or defer to infinity. Consider a rational expression involving factors of the form \(x + 2\) and \(x - 4\).
03

Establish the inequality

Combine these factors into a rational function, for instance \[ \frac{1}{(x+2)(x-4)} \leq 0 \]. This selection ensures that the rational function will be zero or negative in the given intervals.
04

Verification

Evaluate the sign of \ \frac{1}{(x+2)(x-4)} \ for the intervals \(-鈭�, -2] \cup [4, 鈭�)\) to confirm the critical points included in the rational inequality solution set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solution Set
The solution set of an inequality represents all the possible values of the variable that satisfy the inequality. In this problem, the solution set is given as \( (-\rightarrow , -2] \cup [4, \rightarrow )\). This notation means that the values of x can be from negative infinity up to -2 (including -2), and from 4 to positive infinity (including 4). Visualizing this on a number line can help better understand the intervals.
Critical Points
Critical points are essential in defining where the behavior of a function or inequality changes. Here, the critical points are -2 and 4. These values mark the boundaries where the solution set splits the number line into different intervals. We analyze the sign of the rational expression in these intervals to determine where it satisfies the inequality.
Rational Expression
A rational expression is a fraction where both the numerator and the denominator are polynomials. In our case, we use the rational expression \( \frac{1}{(x+2)(x-4)} \) to model the inequality. This choice is because we need the factors (x + 2) and (x - 4) to help create intervals that correspond to our critical points -2 and 4.
Inequality Verification
Verification ensures our rational inequality accurately represents the desired solution set. We evaluate \( \frac{1}{(x+2)(x-4)} \leq 0 \) in different intervals determined by our critical points (-2 and 4). By checking the sign of the expression in these intervals, you can confirm that the inequality holds true within \( -鈭� \leq -2\) and \( 4 \leq 鈭瀄).

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