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Chapter 2: Problem 9

If \(S\) is the set of all real \(x\) and such that \(\frac{2 x-1}{2 x^{3}+3 x^{2}+x}\) is positive, then \(S\) contains (A) \(\left(-\infty,-\frac{3}{2}\right)\) (B) \(\left(-\frac{3}{2},-\frac{1}{4}\right)\) (C) \(\left(-\frac{1}{4}, \frac{1}{2}\right)\) (D) \(\left(\frac{1}{2}, 3\right)\)

Short Answer

Expert verified
S contains options A and D.

Step by step solution

01

Identify the Signs of the Factors

To determine where the expression \(\frac{2x-1}{2x^3+3x^2+x}\) is positive, analyze the numerator \(2x-1\) and the denominator \(2x^3+3x^2+x\). Check for values of \(x\) that make the expression positive or negative.
02

Factoring the Denominator

Factor the denominator: \(2x^3 + 3x^2 + x = x(2x^2 + 3x + 1)\). Find the roots of \(2x^2 + 3x + 1\) using the quadratic formula, which gives \(x = -1\) and \(x = -\frac{1}{2}\). Thus, the denominator is zero at \(x = 0, -1, -\frac{1}{2}\), creating potential sign changes.
03

Determine Numerator's Sign

The numerator \(2x-1\) is zero at \(x = \frac{1}{2}\). It is negative when \(x < \frac{1}{2}\) and positive when \(x > \frac{1}{2}\).
04

Analyze Intervals Between Roots

Divide the real number line into intervals based on points where the expression is zero: \((-\infty, -1), (-1, -\frac{1}{2}), (-\frac{1}{2}, 0), (0, \frac{1}{2}), (\frac{1}{2}, \infty)\). Evaluate the sign of the expression within each interval by picking test points.
05

Evaluate Expression's Sign in Each Interval

- In \((-\infty, -1)\), pick \(x = -2\): both factors are negative, the expression is positive.- In \((-1, -\frac{1}{2})\), pick \(x = -\frac{3}{4}\): numerator is negative, denominator is positive, the expression is negative.- In \((-\frac{1}{2}, 0)\), pick \(x = -\frac{1}{4}\): both factors are negative, the expression is positive.- In \((0, \frac{1}{2})\), pick \(x = \frac{1}{4}\): numerator is negative, denominator is positive, the expression is negative.- In \((\frac{1}{2}, \infty)\), pick \(x = 1\): both factors are positive, the expression is positive.
06

Identify Positive Intervals

The expression is positive in the intervals \((-\infty, -1)\), \((-\frac{1}{2}, 0)\) and \((\frac{1}{2}, \infty)\). Match these with the given options to check which intervals are entirely contained.

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Key Concepts

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

Real Numbers
Real numbers include every kind of number you can visualize on a number line. This includes integers like -3, 0, or 4, and fractions like -1/2 or 2.5. Essentially, any number that doesn't have an imaginary component (like the square root of a negative number) falls under real numbers. When you work with inequalities, you're often using real numbers as these variables can continuously vary.

Here is a quick breakdown of real numbers:
  • **Rational Numbers**: Numbers that can be expressed as a fraction, such as 1/2, -3/4, or 5.
  • **Irrational Numbers**: Numbers that cannot be expressed as fractions, like \( ext{\(\pi\)}\) or the square root of 2.
  • **Integers**: Whole numbers that can be positive, negative, or zero.
Understanding real numbers helps you grasp the foundational concepts of algebra and coordinate geometry, making it easier to solve inequalities.
Interval Notation
Interval notation is a shorthand used to describe a range of numbers on the real number line. It's a convenient way to communicate precisely where numbers fall within the boundaries. When you're working with inequalities, interval notation helps you express the solution set clearly.

Here's how it breaks down:
  • **Closed Intervals [a, b]**: Include both endpoints, meaning all numbers between a and b are included.
  • **Open Intervals (a, b)**: Do not include the endpoints. They only contain numbers strictly between a and b.
  • **Half-Open Intervals [a, b) or (a, b]**: Include one of the endpoints鈥攅ither a or b鈥攂ut not both.
For example, \( (0, 3] \) means all numbers greater than 0 up to and including 3. This notation is essential to understanding the solution set for inequalities.
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its simpler components, which are usually polynomials of lower degree. This process helps simplify equations, making it easier to identify solutions, especially in solving inequalities.

When you factor a polynomial, you're looking for terms that can be regrouped into a product of terms:
  • **Factor out Common Terms**: First, look for common terms that can be factored out, such as pulling out an 'x' in the expression \(2x^3 + 3x^2 + x\).
  • **Use Special Factoring Formulas**: Apply formulas for special cases like difference of squares or perfect squares. These make certain expressions easier to factor.
  • **Quadratic Formula for Non-Trivial Cases**: When a polynomial cannot be factored easily, use the quadratic formula to find roots. For example, use it on \(2x^2 + 3x + 1\) to find the roots \(x = -1\) and \(x = -\frac{1}{2}\).
Mastery of polynomial factorization improves your ability to identify intervals where an expression is positive or negative, aiding in solving polynomial inequalities.

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Most popular questions from this chapter

If \(q^{2}-4 p r=0, p>0\), then the domain of the function \(f(x)=\log \left[p x^{3}+(p+q) x^{2}+(q+r) x+r\right]\) is (A) \(R-\left\\{-\frac{q}{2 p}\right\\}\) (B) \(R-\left[(-\infty,-1] \cup\left\\{-\frac{q}{2 p}\right\\}\right]\) (C) \(R-\left[(-\infty,-1) \cap\left\\{-\frac{q}{2 p}\right\\}\right]\) (D) None of these

If \(f:(0, \pi) \rightarrow R\) be defined by \(f(x)=\sum_{k=1}^{n}([1+\sin k x])\), where \([x]\) denotes the integral part of \(x\), then the range of \(f(x)\) is (A) \(\\{n-1, n+1\\}\) (B) \(\\{n-1, n, n+1\\}\) (C) \(\\{n, n+1\\}\) (D) None of these

The range of the function \(y=\sin ^{-1}\left[x^{2}+\frac{1}{2}\right]+\cos ^{-1}\left[x^{2}-\frac{1}{2}\right]\), where \([\cdot]\) denotes the integral part, is (A) \((0, \pi)\) (B) \([0, \pi]\) (C) \(\\{\pi\\}\) (D) \(\\{0, \pi\\}\)

The values of \(x\) for which the functions \(f(x)=x-3\) and \(\phi(x)=4-x\) satisfy the inequality \(|f(x)+\phi(x)|<\) \(|f(x)|+|\phi(x)|\) are (A) \([3,4]\) (B) \((-\infty, \infty)\) (C) \((-\infty, \infty)-[3,4]\) (D) None of these

The function \(f:(-\infty,-1] \rightarrow\left(0, e^{5}\right]\) defined by, \(f(x)=e^{x^{3}-3 x+2}\) is (A) Many one and onto (B) Many one and into (C) One-one and onto (D) One-one and into
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