Solve 1/3(3x-6)^3+4<13
1/3(3x-6)^3+4 < 13.
1/3(3x-6)^2(3x-6)+4 < 13,
1/3(9x^2-36x+36)(3x-6)+4 < 13,
(3x^2-12x+12)(3x-6)+4 < 13 ,
3(x^2-4x+4)(3x-6)+4 < 13,
3(x^2-4x+4)3(x-2)+4 < 13,
9(x^2-4x+4)(x-2)+4 < 13,
9(x^3-2x^2-4x^2+8x+4x-8)+4 < 13,
9(x^3-6x^2+12x-8)+4 < 13,
9(x^3-6x^2+12x-8)-9 < 0,
Divide both sides by 9:
(x^3-6x^2+12x-8)-1 < 0,
Using synthetic division:
(x-2)(x^2-4x+4)-1 < 0,
(x-2)(x-2)(x-2) < 1,
x-2 < 1,
X < 3.
To solve the inequality 1/3(3x-6)^3+4<13, we will follow these steps:
Step 1: Simplify the expression inside the parentheses.
Start by dividing 3x by 3, and -6 by 3:
(3x-6)^3 + 4 < 13
(x-2)^3 + 4 < 13
Step 2: Expand the expression.
(x-2)^3 can be expanded using the formula for the cube of a binomial:
(x-2)(x-2)(x-2)
(x^2 - 4x + 4)(x-2)
x^3 - 4x^2 + 4x - 2x^2 + 8x - 8
x^3 - 6x^2 + 12x - 8 + 4 < 13
x^3 - 6x^2 + 12x - 4 < 13
Step 3: Combine like terms.
x^3 - 6x^2 + 12x - 4 - 13 < 0
x^3 - 6x^2 + 12x - 17 < 0
Step 4: Solve the inequality.
To solve the inequality, we can use a number line or algebraic methods such as factoring or the rational roots theorem. Since factoring or finding the exact roots might be complicated, we can use a graphing calculator or a graphing software to find the approximate solutions.
By graphing the function f(x) = x^3 - 6x^2 + 12x - 17, we can see that the function is below the x-axis between approximately x ≈ 0.85 and x ≈ 1.54.
Therefore, the solution to the inequality 1/3(3x-6)^3+4<13 is approximately 0.85 < x < 1.54.
To solve the inequality 1/3(3x-6)^3 + 4 < 13, follow these steps:
Step 1: Subtract 4 from both sides of the inequality to isolate the expression.
1/3(3x-6)^3 < 9
Step 2: Multiply both sides of the inequality by 3 to eliminate the fraction.
(3/3)(1/3)(3x-6)^3 < 9 * 3
(3x-6)^3 < 27
Step 3: Take the cube root on both sides of the inequality to remove the cubic exponent.
∛[(3x-6)^3] < ∛27
3x-6 < 3
Step 4: Add 6 to both sides of the inequality.
3x - 6 + 6 < 3 + 6
3x < 9
Step 5: Divide both sides of the inequality by 3.
(1/3)(3x) < (9/3)
x < 3
Thus, the solution to the inequality 1/3(3x-6)^3 + 4 < 13 is x < 3.