Question

The set of values for which \(x^{3}+1 \geq x^{2}+x\) is : (a) \(x \geq 0\) (b) \(x \leq 0\) (c) \(x \geq-1\) (d) \(-1 \leq x \leq 1\)

Short answer

Answer: (c) \(x \geq -1\)

Step-by-step solution

Simplify the inequality

Subtract \(x^2 + x\) from both sides of the inequality to combine terms: \(x^3 + 1 - x^2 - x \geq 0\) This simplifies to: \(x^3 - x^2 - x + 1\geq 0\)

Factor the expression

We will try to factor the left-side expression of the inequality: $$ x^3 - x^2 - x + 1\geq 0 $$ Notice that the expression has a common factor of x: $$ x(x^2 - x - 1) + 1\geq 0 $$ Now, we need to further factor the expression inside the parentheses. This is a quadratic expression, so let's try to factor it using the quadratic formula: $$ x = \frac{-B \pm \sqrt{B^2-4AC}}{2A} $$ Where \(A=-1\), \(B=-1\), and \(C=1\): $$ x = \frac{-(-1) \pm \sqrt{(-1)^2-4(-1)(1)}}{2(-1)} $$ However, the discriminant becomes negative: $$ (-1)^2 - 4(-1)(1) = 1 + 4 = 5 $$ Since the discriminant is not a perfect square, we cannot factor this expression further.

Analyze the inequality using a number line and test points

Now that we have the inequality in its simplest form: $$ x(x^2 - x - 1) + 1 \geq 0 $$ We can use a number line and test points to find the intervals where the inequality holds true. Divide the number line into three regions based on the critical points: 1. \(x < -1\) 2. \(-1\leq x < 0\) 3. \(x \geq 0\) Now, we will test the inequality with one chosen point from each interval.

Test interval 1

Choose a test point from the interval \(x < -1\), for example, \(x = -2\). Plug this value into the inequality: $$ x( x^2 - x - 1) + 1 = (-2)((-2)^2 - (-2) - 1) + 1 = -3 $$ Since \(-3\) is not greater than or equal to \(0\), the inequality does not hold true for this interval.

Test interval 2

Choose a test point from the interval \(-1\leq x < 0\), for example, \(x = -\frac{1}{2}\). Plug this value into the inequality: $$ x( x^2 - x - 1) + 1 = \left(-\frac{1}{2}\right)\left(\left(-\frac{1}{2}\right)^2 - \left(-\frac{1}{2}\right) - 1\right) + 1 = \frac{3}{4} $$ Since \(\frac{3}{4}\) is greater than or equal to \(0\), the inequality holds true for this interval.

Test interval 3

Choose a test point from the interval \(x \geq 0\), for example, \(x = 1\). Plug this value into the inequality: $$ x( x^2 - x - 1) + 1 = (1)((1)^2 - (1) - 1) + 1 = 0 $$ Since \(0\) is greater than or equal to \(0\), the inequality holds true for this interval as well. #Conclusion# From our testing, we found that the inequality holds true for the intervals \(-1 \le x < 0\) and \(x \ge 0\). Combining these intervals, we get the solution as \(-1 \le x\). Comparing this with our given choices, it matches choice (c), so the correct answer is: (c) \(x \geq -1\)

Key concepts

Quadratic Formula

The quadratic formula is a crucial tool in solving quadratic equations, which takes the standard form \(ax^2 + bx + c = 0\). This formula allows you to find the roots of any quadratic equation and is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(a\), \(b\), and \(c\) are the coefficients of the polynomial equation. The part under the square root, \(b^2 - 4ac\), is known as the discriminant. It determines the nature of the roots of the quadratic equation. The discriminant can tell us:

• If it is positive, there are two distinct real roots.
• If it is zero, there are exactly one or two equal real roots.
• If it is negative, the roots are complex or imaginary numbers.

In the exercise, the discriminant \((-1)^2 - 4(-1)(1)\) equals \(5\). Since this is positive but not a perfect square, the roots are real and irrational, meaning the expression does not factor neatly into real numbers.

Polynomials

Polynomials are algebraic expressions that involve sums of powers of variables with coefficients. The general form of a polynomial in one variable \(x\) is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(n\) is a non-negative integer, and \(a_n, a_{n-1}, \ldots, a_0\) are constants.
Polynomials can be classified by their degree, which is the highest power of the variable in the expression. A cubic polynomial, like the one in the exercise \(x^3 - x^2 - x + 1\), has a degree of 3.
Solving polynomial inequalities, like the one in the exercise, often involves:

• Simplifying and factoring the polynomial expression.
• Using the roots or critical points to test intervals on the number line.

Understanding the properties of polynomials helps you approach inequalities methodically and apply calculus-based or algebraic techniques to factor or simplify them, whenever possible.

Number Line Analysis

Number line analysis is a graphical approach to solve inequalities by visually determining where they hold true. By understanding different ranges on the number line, it becomes easier to see where expressions are positive, negative, or zero.
To perform number line analysis:

• Identify critical points (roots of the equation or where expressions may change sign).
• Divide the number line into segments based on these critical points.
• Choose test points from each segment to determine whether the inequality is satisfied.

In the exercise, the number line is divided around the critical points \(-1\), \(0\), and positive infinity. Testing regions like \(x < -1\), \(-1 \leq x < 0\), and \(x \geq 0\) with specific values helps determine which intervals satisfy the inequality \(x(x^2 - x - 1) + 1 \geq 0\). The correct segments, where the inequality holds, indicate the solution range.

Mathematical Reasoning

Mathematical reasoning is the logical thought process used to solve problems and justify steps in solutions. This skill is essential in breaking down complex tasks, like inequality solving, into manageable steps.
In mathematical reasoning:

• Understanding the problem thoroughly by defining what is known and what needs to be determined.
• Breaking the problem into smaller parts to simplify and solve them separately.
• Using known concepts and logic to draw conclusions and validate the steps taken.

In the exercise, after simplifying the polynomial inequality, reasoning is used to decide how best to solve it. Recognizing that factoring isn't straightforward due to the discriminant, the process of testing intervals on the number line is chosen. This strategic approach helps validate which intervals satisfy the original inequality, thus, demonstrating effective mathematical reasoning.