Question

The inequality \(x^{4}+1<-5\) has no solution. Explain why.

Short answer

The inequality has no solution because the left side is always greater than or equal to 1, which is never less than -5.

Step-by-step solution

- Recognize the inequality

The given inequality is \(x^{4} + 1 < -5\).

- Understand the mathematical properties

Recall that any number raised to the fourth power, \(x^{4}\), will always be non-negative (≥ 0).

- Analyze the inequality with non-negative values

Since \(x^{4} \geq 0\), the smallest value \(x^{4} + 1\) can take is 1 (when \(x=0\)). Therefore, \(x^{4} + 1\) is always greater than or equal to 1.

- Compare with the given inequality

The inequality \(x^{4} + 1 < -5\) suggests that \(x^{4} + 1\) should be less than -5. However, as established in Step 3, the smallest value of \(x^{4} + 1\) is 1, which is always greater than -5.

- Conclude there is no solution

Since \(x^{4} + 1\) can never be less than -5, the inequality \(x^{4} + 1 < -5\) has no solution.

Key concepts

non-negative values

When solving polynomial inequalities, understanding non-negative values is crucial. Here, we focus on the expression involving monomials raised to a power. Specifically, in the inequality you are dealing with, \(x^{4} + 1 < -5\), it's important to acknowledge a key concept: any even power of a real number is non-negative.
For instance, \(x^{4}\) signifies that no matter what value x assumes, whether positive, negative, or zero, \(x^{4}\) will always result in a value greater than or equal to 0. Considering this critical point, the minimum value \[x^{4}\] can take is 0. So, the smallest value of \(x^{4} + 1\) must be 1. This understanding simplifies resolving inequalities since you can infer that certain expressions involving \[x^{4}\] will always be non-negative.

inequality analysis

Analyzing inequalities often involves comparing the expressions to fixed values. In the given inequality \(x^{4} + 1 < -5\), it’s helpful to break it down by comparing the terms on the left-hand side with the value on the right-hand side. To see why there's no solution, follow a step-by-step examination:
Start by acknowledging that \(x^{4} + 1\) cannot be less than -5 because \(x^{4} \) is always non-negative. Understanding the bounds for polynomial expressions allows us to confidently determine that \(x^{4} + 1 \) is always at least 1. In a mathematically unrealistic situation wherein the expression on the left must be less than -5, it directly implies there's no value of \(x\) that makes the inequality true.
Thus, comparing the theoretical minimum values reveals inconsistencies, making it clear that the inequality has no solution.

polynomials

Polynomials are algebraic expressions consisting of variables and coefficients that are related by operations of addition, subtraction, and multiplication. The polynomial of interest in our context is \(x^{4} + 1\). Understanding the behavior of polynomial terms, especially higher-degree terms like x raised to the fourth power, is fundamental.
When dealing with polynomials, recognizing the nature and degree can help predict their properties. A polynomial’s degree impacts its shape and range significantly. Here, \(x^{4}\) as an even-powered term ensures non-negative values, guiding us on how expressions involving such terms behave. Therefore, realizing that the smallest term in a polynomial including \(x^{4}+1\) is always 1 when \(x=0\), shows the polynomial cannot yield values small enough to satisfy \(x^{4} + 1 < -5\).
Polynomials' properties provide essential insights when solving related inequalities and interpreting the feasibility of obtaining solutions.