Question

(3x + 1)(x - 2) < 6;​

Answer 1

...............................

Answer 2

To solve the inequality (3x + 1)(x - 2) < 6, follow these steps:

1. Expand the left side of the inequality:

   (3x + 1)(x - 2) < 6 3x^2 - 6x + x - 2 < 6

2. Combine like terms:

   3x^2 - 5x - 2 < 6

3. Move all terms to one side of the inequality to set it equal to zero:

   3x^2 - 5x - 2 - 6 < 0

4. Simplify:

   3x^2 - 5x - 8 < 0

Now, we need to find the values of x that satisfy this inequality. To do that, we can use the quadratic formula:

The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a)

In our case, a = 3, b = -5, and c = -8. Plug these values into the quadratic formula:

x = (-(-5) ± √((-5)² - 4(3)(-8))) / (2(3))

x = (5 ± √(25 + 96)) / 6

x = (5 ± √121) / 6

x = (5 ± 11) / 6

Now, we have two possible values for x:

1. x = (5 + 11) / 6 = 16 / 6 = 8/3
2. x = (5 - 11) / 6 = -6 / 6 = -1

So, the solutions to the inequality 3x^2 - 5x - 8 < 0 are x < -1 and x > 8/3. You can represent this solution on a number line or in interval notation:

In interval notation: x ∈ (-∞, -1) ∪ (8/3, ∞)

In a number line:

scssCopy code
<-----|---(-1)---|---(8/3)---|----->

These are the values of x that satisfy the given inequality.