Solving $2x^3 + 4x \geq 4x^2 + 6x - 4$ A Step-by-Step Guide

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In this article, we will delve into solving the cubic inequality $2x^3 + 4x \geq 4x^2 + 6x - 4$. Inequalities of this nature can be challenging, but by employing algebraic manipulation, factoring techniques, and careful analysis of intervals, we can arrive at the solution set. This comprehensive guide will walk you through each step, providing explanations and insights to help you understand the process thoroughly. We'll start by rearranging the inequality, factoring the resulting polynomial, identifying critical points, and finally, expressing the solution in interval notation. This process not only solves the specific inequality but also equips you with valuable skills for tackling similar problems in mathematics. Understanding how to solve polynomial inequalities is crucial in various fields, including calculus, optimization problems, and real-world applications where constraints and limitations need to be modeled mathematically. So, let's dive in and unravel the solution to this interesting problem.

The problem we aim to solve is the cubic inequality:

$2x^3 + 4x \geq 4x^2 + 6x - 4$

To solve this inequality, we need to find the values of $x$ that satisfy the given condition. This involves a series of algebraic steps, including rearranging the inequality to have zero on one side, factoring the polynomial, and identifying the intervals where the inequality holds true. Solving inequalities is a fundamental skill in algebra and calculus, with applications in various fields, such as optimization, modeling, and analysis of real-world scenarios. Inequalities help us define ranges and constraints, making them essential tools in problem-solving. In this specific case, we are dealing with a cubic inequality, which means the highest power of $x$ is 3. Solving cubic inequalities often involves finding the roots of the corresponding cubic equation and then testing intervals to determine where the inequality is satisfied. This process can be more complex than solving linear or quadratic inequalities, but with a systematic approach, we can break it down into manageable steps and arrive at the correct solution. The ultimate goal is to express the solution set in interval notation, which provides a concise and clear representation of all the values of $x$ that satisfy the inequality.

Step 1: Rearrange the Inequality

To begin, we need to rearrange the inequality so that one side is zero. This makes it easier to factor and analyze the polynomial. Subtract $4x^2 + 6x - 4$ from both sides of the inequality:

$2x^3 + 4x - (4x^2 + 6x - 4) \geq 0$

Simplify the expression:

$2x^3 - 4x^2 + 4x - 6x + 4 \geq 0$

Combine like terms:

$2x^3 - 4x^2 - 2x + 4 \geq 0$

This rearranged inequality sets the stage for factoring, which is the next crucial step in solving the problem. By moving all terms to one side, we can now focus on finding the roots of the cubic polynomial and determining the intervals where the inequality holds. Rearranging the inequality is a common practice in solving polynomial inequalities, as it allows us to analyze the behavior of the polynomial function relative to zero. This step is not only important for solving this specific problem but is also a fundamental technique in various mathematical contexts. The simplified form of the inequality, $2x^3 - 4x^2 - 2x + 4 \geq 0$, is now ready for further manipulation and factorization.

Step 2: Factor the Polynomial

Now, let's factor the polynomial $2x^3 - 4x^2 - 2x + 4$. First, we can factor out a common factor of 2:

$2(x^3 - 2x^2 - x + 2) \geq 0$

Next, we can try factoring by grouping. Group the terms in pairs:

$2[(x^3 - 2x^2) + (-x + 2)] \geq 0$

Factor out the greatest common factor from each group:

$2[x^2(x - 2) - 1(x - 2)] \geq 0$

Now, we can factor out the common binomial factor $(x - 2)$:

$2(x - 2)(x^2 - 1) \geq 0$

Finally, factor the difference of squares $(x^2 - 1)$:

$2(x - 2)(x - 1)(x + 1) \geq 0$

Factoring the polynomial is a critical step in solving polynomial inequalities. It allows us to identify the roots of the polynomial, which are the values of $x$ that make the polynomial equal to zero. These roots serve as the critical points that divide the number line into intervals. By factoring, we transform the cubic polynomial into a product of linear factors, making it easier to analyze its sign in different intervals. The factored form, $2(x - 2)(x - 1)(x + 1) \geq 0$, reveals the roots of the polynomial as $x = -1$, $x = 1$, and $x = 2$. These roots are the key to determining the solution set of the inequality. Factoring by grouping, as demonstrated in this step, is a powerful technique that can be applied to various polynomial expressions.

Step 3: Identify Critical Points

The critical points are the values of $x$ that make the polynomial equal to zero. From the factored form $2(x - 2)(x - 1)(x + 1) \geq 0$, we can identify the critical points as:

• $x = -1$
• $x = 1$
• $x = 2$

These critical points divide the number line into intervals, which we will analyze in the next step to determine where the inequality holds true. Identifying critical points is a fundamental step in solving polynomial inequalities. These points are the roots of the polynomial and represent the values of $x$ where the polynomial changes its sign. The critical points effectively partition the number line into distinct intervals, within each of which the polynomial's sign remains constant. By determining the sign of the polynomial in each interval, we can identify the regions where the inequality is satisfied. In this case, the critical points $-1$, $1$, and $2$ divide the number line into four intervals: $(-\infty, -1)$, $(-1, 1)$, $(1, 2)$, and $(2, \infty)$. Each of these intervals will be tested to see if the polynomial is greater than or equal to zero. Understanding the concept of critical points is essential for solving a wide range of inequality problems.

Step 4: Analyze Intervals

Now, we will analyze the intervals created by the critical points to determine where the inequality $2(x - 2)(x - 1)(x + 1) \geq 0$ holds true. The intervals are:

1. $(-\infty, -1)$
2. $(-1, 1)$
3. $(1, 2)$
4. $(2, \infty)$

We will pick a test value within each interval and plug it into the factored inequality to determine the sign of the polynomial in that interval.

Interval 1: $(-\infty, -1)$

Let's pick $x = -2$ as a test value:

$2(-2 - 2)(-2 - 1)(-2 + 1) = 2(-4)(-3)(-1) = -24$

Since $-24 < 0$, the inequality is not satisfied in this interval.

Interval 2: $(-1, 1)$

Let's pick $x = 0$ as a test value:

$2(0 - 2)(0 - 1)(0 + 1) = 2(-2)(-1)(1) = 4$

Since $4 > 0$, the inequality is satisfied in this interval.

Interval 3: $(1, 2)$

Let's pick $x = 1.5$ as a test value:

$2(1.5 - 2)(1.5 - 1)(1.5 + 1) = 2(-0.5)(0.5)(2.5) = -1.25$

Since $-1.25 < 0$, the inequality is not satisfied in this interval.

Interval 4: $(2, \infty)$

Let's pick $x = 3$ as a test value:

$2(3 - 2)(3 - 1)(3 + 1) = 2(1)(2)(4) = 16$

Since $16 > 0$, the inequality is satisfied in this interval.

Analyzing intervals is a crucial step in solving inequalities. By dividing the number line into intervals based on the critical points, we can systematically determine where the polynomial is positive, negative, or zero. This process involves selecting a test value within each interval and evaluating the polynomial at that value. The sign of the polynomial at the test value indicates the sign of the polynomial throughout the entire interval. This method works because the sign of a polynomial can only change at its roots (critical points). In this step, we tested each interval and found that the inequality $2(x - 2)(x - 1)(x + 1) \geq 0$ is satisfied in the intervals $(-1, 1)$ and $(2, \infty)$. This analysis provides the foundation for expressing the solution set in interval notation. The careful selection of test values and the accurate evaluation of the polynomial are essential for obtaining the correct solution.

Step 5: Express the Solution in Interval Notation

Now, we can express the solution in interval notation. The inequality is satisfied in the intervals $(-1, 1)$ and $(2, \infty)$. We also need to include the critical points themselves since the inequality is non-strict (i.e., $\geq$ includes equality). Therefore, the solution in interval notation is:

$[-1, 1] \cup [2, \infty)$

This interval notation represents all the values of $x$ that satisfy the original inequality. The square brackets indicate that the endpoints are included in the solution set. Expressing the solution in interval notation is the final step in solving the inequality. This notation provides a concise and clear representation of all the values of $x$ that satisfy the given condition. The union symbol $\cup$ indicates that the solution set consists of two disjoint intervals. In this case, the solution includes all values of $x$ between $-1$ and $1$ (inclusive), as well as all values of $x$ greater than or equal to $2$. The use of square brackets to include the endpoints $-1$, $1$, and $2$ is crucial because the original inequality includes the equality case ($\geq$). Interval notation is a standard way of representing solutions to inequalities and is widely used in mathematics and related fields. The solution $[-1, 1] \cup [2, \infty)$ provides a complete and accurate answer to the problem.

The solution to the inequality $2x^3 + 4x \geq 4x^2 + 6x - 4$ in interval notation is:

$[-1, 1] \cup [2, \infty)$

This final answer encapsulates the entire process of solving the inequality, from rearranging and factoring the polynomial to analyzing intervals and expressing the solution in interval notation. The solution set represents all real numbers $x$ that satisfy the given inequality. This comprehensive solution not only answers the specific problem but also demonstrates the methodology for solving similar cubic inequalities. Understanding the steps involved, such as factoring, identifying critical points, and testing intervals, is essential for mastering inequality problems. The final answer, $[-1, 1] \cup [2, \infty)$, is the culmination of these steps and provides a clear and concise representation of the solution set. This result can be used in various applications, such as optimization problems, where finding the range of values that satisfy certain conditions is crucial. The ability to solve inequalities is a valuable skill in mathematics and related fields.