Solve the quadratic inequality 2x^2 + x - 3 > 0

is this correct? I'm not sure what a solved quadratic inequality looks like.
2x^2 + x > 3

2x^2 + x - 3 > 0

(2x + 3)(x - 1) > 0

which tells me that the x-intercepts of the parabola
y = 2x^2 + x - 3 are -3/2 and 1 and it opens upwards.

So for what values is the function above the x-axis ?

That is a parabola

for very large |x| , y is +
so it opens up (holds water)
so find the roots and any x< the lower one or > the upper one satisfies the condition
2 x^2 + x - 3 = 0
(2x-3)(x+1) = 0
any x < -1 is good
any x > 3/2 is good
in between , no good

woops sorry

(2x+3)(x-1) =0 for roots
x<-3/2 or x > 1

To solve the quadratic inequality 2x^2 + x - 3 > 0, we need to find the values of x that satisfy the inequality.

1. Start by setting the quadratic expression equal to zero to find the critical points:
2x^2 + x - 3 = 0

2. Factor the quadratic equation, or use the quadratic formula, to solve for x:
(2x - 1)(x + 3) = 0

Setting each factor equal to zero:
2x - 1 = 0 --> x = 1/2
x + 3 = 0 --> x = -3

So, the critical points are x = 1/2 and x = -3.

3. We will now use these critical points to split the number line into three intervals to determine the sign of the quadratic expression in each interval.

Interval 1: x < -3
Choose a value for x less than -3, such as x = -4:
Substitute x = -4 into the inequality:
2(-4)^2 + (-4) - 3 > 0
32 - 4 - 3 > 0
25 > 0

The expression 2x^2 + x - 3 is positive for x < -3.

Interval 2: -3 < x < 1/2
Choose a value for x between -3 and 1/2, such as x = 0:
Substitute x = 0 into the inequality:
2(0)^2 + (0) - 3 > 0
0 - 3 > 0
-3 > 0

The expression 2x^2 + x - 3 is negative for -3 < x < 1/2.

Interval 3: x > 1/2
Choose a value for x greater than 1/2, such as x = 1:
Substitute x = 1 into the inequality:
2(1)^2 + (1) - 3 > 0
2 + 1 - 3 > 0
0 > 0

The expression 2x^2 + x - 3 is negative for x > 1/2.

4. Combining the information from the intervals, we can conclude that the inequality 2x^2 + x - 3 > 0 is satisfied when x < -3 or x > 1/2.

Therefore, the correct form of the quadratic inequality is: x < -3 OR x > 1/2.