Math Problem Statement
https://latex.codecogs.com/svg.image?&space;x^{3}+1\leq&space;0
Solution
To solve the inequality , let's proceed step by step:
1. Simplify the inequality:
Subtract 1 from both sides:
2. Solve for :
The solution involves finding the cube root:
The cube root of is , so:
3. Solution:
The solution set for the inequality is:
This represents all values of that are less than or equal to .
4. Graphical Representation:
The function has a root at , and the cubic function decreases for . The inequality is satisfied in this region.
Do you want a graph or further details?
Related Questions:
- What is the general approach for solving polynomial inequalities?
- How can the graph of a cubic function help in understanding inequalities?
- What is the difference between solving and inequalities?
- How do we verify solutions to inequalities algebraically?
- 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