Math Problem Statement

https://latex.codecogs.com/svg.image?&space;x^{3}+1\leq&space;0

Solution

To solve the inequality x3+10x^3 + 1 \leq 0, let's proceed step by step:


1. Simplify the inequality:

x3+10x^3 + 1 \leq 0

Subtract 1 from both sides: x31x^3 \leq -1


2. Solve for xx:

The solution involves finding the cube root: x13x \leq \sqrt[3]{-1}

The cube root of 1-1 is 1-1, so: x1x \leq -1


3. Solution:

The solution set for the inequality is: x(,1]x \in (-\infty, -1]

This represents all values of xx that are less than or equal to 1-1.


4. Graphical Representation:

The function x3+1x^3 + 1 has a root at x=1x = -1, and the cubic function decreases for x<1x < -1. The inequality x3+10x^3 + 1 \leq 0 is satisfied in this region.


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Related Questions:

  1. What is the general approach for solving polynomial inequalities?
  2. How can the graph of a cubic function help in understanding inequalities?
  3. What is the difference between solving \leq and << inequalities?
  4. How do we verify solutions to inequalities algebraically?
  5. Can we apply similar methods to higher-degree polynomials?

Tip:

Always check for critical points and analyze the sign of the function in the intervals they create.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Cubic Functions

Formulas

x^3 ≤ -1
x ≤ ∛(-1)

Theorems

Properties of Inequalities
Root Analysis for Cubic Functions

Suitable Grade Level

Grades 9-12