Sure, I'll provide a detailed step-by-step solution to the inequality [tex]\( 6x + 9y^3 - 5x^4 - 8 + 6y^5 < 2y + 1 \)[/tex].
### Step 1: Write Down the Given Inequality
The inequality we need to solve is:
[tex]\[ 6x + 9y^3 - 5x^4 - 8 + 6y^5 < 2y + 1 \][/tex]
### Step 2: Bring All Terms to One Side of the Inequality
Subtract [tex]\( 2y + 1 \)[/tex] from both sides to bring all terms to the left side:
[tex]\[ 6x + 9y^3 - 5x^4 - 8 + 6y^5 - 2y - 1 < 0 \][/tex]
### Step 3: Combine Like Terms
We need to simplify by combining like terms:
[tex]\[ 6x + 9y^3 + 6y^5 - 5x^4 - 2y - 9 < 0 \][/tex]
### Step 4: Reorder the Terms
Reorder the terms to present the polynomial in a standard form:
[tex]\[ -5x^4 + 6x + 6y^5 + 9y^3 - 2y - 9 < 0 \][/tex]
### Step 5: Define the Expressions
Identify the left-hand side (lhs) and right-hand side (rhs) of the inequality:
Left-hand side:
[tex]\[ \text{lhs} = -5x^4 + 6x + 6y^5 + 9y^3 - 2y - 9 \][/tex]
Right-hand side:
[tex]\[ \text{rhs} = 0 \][/tex]
### Step 6: Express the Inequality in Standard Form
This step clarifies the inequality in a standard mathematical form:
[tex]\[ -5x^4 + 6x + 6y^5 + 9y^3 - 2y - 9 < 0 \][/tex]
So, to solve the inequality:
[tex]\[ 6x + 9y^3 - 5x^4 - 8 + 6y^5 < 2y + 1 \][/tex]
we have transformed it into:
[tex]\[ -5x^4 + 6x + 6y^5 + 9y^3 - 2y - 9 < 0 \][/tex]
Thus, the given inequality is:
[tex]\[ -5x^4 + 6x + 6y^5 + 9y^3 - 8 < 2y + 1 \][/tex]
and in standard polynomial form:
[tex]\[ -5x^4 + 6x + 6y^5 + 9y^3 - 2y - 9 < 0 \][/tex]
That concludes our detailed step-by-step solution to the inequality.
