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Chapter 17: Problem 48

Use inequalities to solve the given problems. Find an inequality of the form \(a x^{3}+b x<0\) with \(a>0\) for which the solution is \(x<-1\) or \(0

Short Answer

Expert verified
The inequality is \(2x^3 - 2x < 0\).

Step by step solution

01

Understanding the Conditions

The inequality needs to be solved such that the solution set is two intervals: \(x < -1\) and \(0 < x < 1\). This means the polynomial has roots at \(x = -1, 0,\) and potentially some point in between or related to these intervals.
02

Formulating the Polynomial

For the inequality \(ax^3 + bx < 0\) and with the solution conditions given, consider that the roots of the polynomial correspond to the boundary of our intervals. Thus, we choose roots at \(x = -1, 0,\) and \(x = 1\). The function should change sign at these points.
03

Constructing the Polynomial

With roots at \(x = -1\), \(0\), and \(1\), the polynomial can be written temporarily as \((x + 1)x(x - 1)\). This transforms into \(x^3 - x\). However, because we need an inequality \(ax^3 + bx < 0\) where \(a > 0\), scale by a positive constant to form: \(2x^3 - 2x < 0\). This maintains \(a > 0\) while having roots at the required points.
04

Validating the Polynomial Against Conditions

For the polynomial \(2x^3 - 2x < 0\), evaluate in the intervals: \(x < -1\), \(0 < x < 1\), and \(x > 1\). These evaluations align with the solution set: at \(x < -1\) and \(0 < x < 1\), the polynomial should be negative. Testing a point in \(x < -1\), say \(x = -2\), gives \(2(-2)^3 - 2(-2) = -16 + 4 = -12 < 0\); for \(0 < x < 1\), say \(x = 0.5\), gives \(2(0.5)^3 - 2(0.5) = 0.25 - 1 = -0.75 < 0\).

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Key Concepts

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

Polynomial Inequalities
A polynomial inequality is a mathematical statement that involves a polynomial expression and an inequality sign. Instead of equating the polynomial to a value, we compare it. For example, in the inequality \( ax^3 + bx < 0 \), our aim is to find the values of \( x \) that make this expression true. When solving polynomial inequalities, it's important to:
  • Identify and understand the degree of the polynomial, as it affects the number of roots and the behavior of the polynomial function.
  • Determine the intervals where the polynomial is greater than or less than zero, typically by testing points within these intervals.
  • Consider how the sign of the leading coefficient (\(a\) in this case) impacts the direction of the inequality's graph.
By spotting where the polynomial crosses the x-axis, or changes sign, we can uncover the solutions for the inequality. Solving such inequalities helps us understand the behavior of the polynomial over different regions of the number line.
Roots of Polynomials
Roots of a polynomial are the values of \( x \) that make the polynomial equal to zero. For example, if we have the equation \( ax^3 + bx = 0 \), the task is to find the roots by setting the equation to zero and solving for \( x \). The roots are crucial because they are the points at which the polynomial changes its sign, which is essential in solving inequalities. To find the roots:
  • Factor the polynomial, if possible. For example, for \( x^3 - x \), you can factor out an \( x \), leaving \( x(x^2 - 1) = 0 \).
  • Continue factoring if needed, \( x^2 - 1 \) can be broken down further into \((x - 1)(x + 1)\).
  • The roots are \( x = 0, \ x = 1, \) and \( x = -1 \).
Identifying roots helps in creating intervals on the number line to easily determine where the polynomial is positive or negative, which in effective assists in solving polynomial inequalities.
Interval Notation
Interval notation is a concise way of expressing the set of solutions for inequalities. Instead of listing all numbers one by one, we can use intervals to describe a range of values that satisfy the inequality. Here's how you can interpret interval notation:
  • Brackets \([\ \text{and}\ ]\) are used when a number is included in the set, known as inclusive boundaries (e.g., \([1, 3]\) includes \(1\) and \(3)\).
  • Parentheses \((\ \text{and}\ )\) show that the boundary number is not included, known as exclusive boundaries (e.g., \((0, 1)\) includes all numbers between \(0\) and \(1\) but not \(0\) or \(1\)).
  • Combination of both, like in \((-\infty, -1) \cup (0, 1)\), illustrates multiple intervals indicating points where our polynomial inequality holds true.
Understanding how to read and write using interval notation is critical for interpreting and communicating solutions to polynomial inequalities effectively.

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Most popular questions from this chapter

Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated system of inequalities. One pump can remove wastewater at the rate of 75 gal/min, and a second pump works at the rate of \(45 \mathrm{gal} / \mathrm{min} .\) Graph the possible values of the time (in min) that each of these pumps operates such that together they pump more than 4500 gal.

Answer the given questions by solving the appropriate inequalities. The weight \(w\) (in \(\mathrm{N}\) ) of an object \(h\) meters above the surface of earth is \(w=r^{2} w_{0} /(r+h)^{2},\) where \(r\) is the radius of earth and \(w_{0}\) is the weight of the object at sea level. Given that \(r=6380 \mathrm{km},\) if an object weighs \(200 \mathrm{N}\) at sea level, for what altitudes is its weight less than \(100 \mathrm{N} ?\)

graph the given inequalities on the number line. \(x < 1\) or \(1 < x \leq 4\)

Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated system of inequalities. A telephone company is installing two types of fiber-optic cable in an area. It is estimated that no more than 300 m of type A cable, and at least 200 m but no more than 400 m of type \(B\) cable, are needed. Graph the possible lengths of cable that are needed.

Solve the given inequalities. Graph each solution. $$(2 x-7)(x+1) \leq 4-x(1-2 x)$$
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