Final answer:
By factoring the quadratic equation and analyzing the intervals on a number line, we can determine the solution to be (-3, 1/2).
Step-by-step explanation:
To solve the inequality 2x^2 + 5x - 3 > 0, we can find the roots of the corresponding quadratic equation 2x^2 + 5x - 3 = 0.
Since the coefficient of x^2 is positive, the parabola opens upwards and we need to find the values of x that make the quadratic expression positive.
We can do this by finding the x-intercepts of the quadratic equation and determining the intervals for which the expression is greater than zero.
- First, let's factor the quadratic equation. We have (2x - 1)(x + 3) = 0.
- Setting each factor equal to zero, we get 2x - 1 = 0 and x + 3 = 0.
- Solving for x, we find x = 1/2 and x = -3.
- Plotting these two x-values on a number line, we have:
- In the region x < -3, the expression 2x^2 + 5x - 3 is negative.
- In the region -3 < x < 1/2, the expression is positive.
- In the region x > 1/2, the expression is also negative.
Therefore, the intervals for which the inequality 2x^2 + 5x - 3 > 0 are: (-3, 1/2).