For this inequality

3x^(2) - 2x -1 > or equal 0

I get x=1 , x=-1/3

would the solution set become
{x|1 < x < -1/3}

I wasn't sure about the sign, if it should be greater than or equal to .

3x^2 - 2x - 1 >= 0,

Factor using the A*C Method:
A * C = 3 * -1 = -3 = 1 * -3,
3x^2 + (x -3x) - 1 >= 0,
(3x^2 - 3x) + (x - 1) >= 0,
3x(x - 1) + (x - 1) >= 0,
(x - 1) (3x + 1) >= 0,

x - 1 >= 0,
x >= 1.

3x + 1 >= 0,
3x >= -1,
x >= -1/3.

Solution Set: x >= 1, and x >= -1/3.

This is not a compound inequality; therefore, your last step does
not apply.

To solve the inequality 3x^2 - 2x - 1 ≥ 0, you correctly found the solutions x = 1 and x = -1/3. However, the solution set would actually be {x | x ≤ -1/3 or x ≥ 1}, not {x | 1 < x < -1/3}.

Here's how you can determine the correct solution set:

1. First, find the critical points of the inequality by setting the expression on the left-hand side of the inequality to zero:
3x^2 - 2x - 1 = 0

To factor or solve this quadratic equation, you can use the quadratic formula. Applying the formula, you would get the solutions x = 1 and x = -1/3 (which you already found).

2. The critical points divide the number line into three regions: to the left of -1/3, between -1/3 and 1, and to the right of 1.

3. Now, choose a test value from each region and substitute it into the original inequality to determine the sign of the expression:

- For a value less than -1/3, let's choose x = -2. Substitute x = -2 into 3x^2 - 2x - 1:
3(-2)^2 - 2(-2) - 1 = 12 + 4 - 1 = 15
Since 15 is greater than zero, the expression is positive in this region.

- For a value between -1/3 and 1, let's choose x = 0. Substitute x = 0 into 3x^2 - 2x - 1:
3(0)^2 - 2(0) - 1 = -1
Since -1 is negative, the expression is negative in this region.

- For a value greater than 1, let's choose x = 2. Substitute x = 2 into 3x^2 - 2x - 1:
3(2)^2 - 2(2) - 1 = 11
Since 11 is positive, the expression is positive in this region.

4. Based on the sign analysis from step 3, we can conclude that the expression 3x^2 - 2x - 1 is positive (greater than zero) to the left of -1/3 and to the right of 1. It is negative (less than zero) between -1/3 and 1.

5. Finally, since the inequality is greater than or equal to, the solution set is the combination of the regions where the expression is positive or zero. So the solution set is {x | x ≤ -1/3 or x ≥ 1}.

In summary, the correct solution set for the inequality 3x^2 - 2x - 1 ≥ 0 is {x | x ≤ -1/3 or x ≥ 1}.