Question

Solve for x: 6x1/4(4x+8)>12

Answer 1

To solve for x, we need to isolate it on one side of the inequality sign.

6x^(1/4)(4x+8) > 12

Divide both sides by 6:

x^(1/4)(4x+8) > 2

Divide both sides by 4:

x^(1/4)(x+2) > 1/2

Raise both sides to the 4th power:

x(x+2)^4 > 1/16

Expand the left side:

x(x^4 + 8x^3 + 24x^2 + 32x + 16) > 1/16

Simplify:

x^5 + 8x^4 + 24x^3 + 32x^2 + 16x - 1/16 > 0

This is a fifth-degree polynomial inequality, which can be difficult to solve algebraically. However, we can use numerical methods or a graphing calculator to find the solution.

One possible solution is x > -0.3. We can check this by plugging in a value slightly larger than -0.3, such as x = -0.2:

x^(1/4)(x+2) = (-0.2)^(1/4)(-0.2+2) ≈ 0.83

This is greater than 1/2, so the inequality holds for x = -0.2.

Therefore, the solution to the inequality is x > -0.3.

Answer 2

To solve for x in the inequality 6x^(1/4)(4x + 8) > 12, we can follow these steps:

1. Simplify both sides by dividing both sides by 6: x^(1/4)(4x + 8) > 2

2. Distribute x^(1/4) into (4x + 8): 4x^(5/4) + 8x^(1/4) > 2

3. Simplify by subtracting 2 from both sides: 4x^(5/4) + 8x^(1/4) - 2 > 0

4. Factor out 2 from the left side: 2(2x^(5/4) + 4x^(1/4) - 1) > 0

5. Divide both sides by 2: 2x^(5/4) + 4x^(1/4) - 1 > 0

We can use a sign table to solve this inequality:

| x^(1/4) | x^(5/4) | 2x^(5/4) | 4x^(1/4) - 1 | 2x^(5/4) + 4x^(1/4) - 1 |
|---------|---------|-----------|--------------|---------------------------|
| 0 | 0 | 0 | -1 | -1 |
| 1 | 1 | 2 | 3 | 5 |

From the table, we can see that the expression 2x^(5/4) + 4x^(1/4) - 1 is greater than 0 when x is either less than 0 or greater than 1. Therefore, the solution to the inequality 6x^(1/4)(4x + 8) > 12 is x < 0 or x > 1.