The inequality shown in the image is:

$2x^2 < x + 6$

Let's solve it step by step:

Step 1: Rewrite the inequality in standard form

Move all terms to one side of the inequality: $2x^2 - x - 6 < 0$

Step 2: Solve the corresponding equation

First, solve the related quadratic equation: $2x^2 - x - 6 = 0$ Using the quadratic formula: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-6)}}{2(2)}$ $x = \frac{1 \pm \sqrt{1 + 48}}{4}$ $x = \frac{1 \pm \sqrt{49}}{4}$ $x = \frac{1 \pm 7}{4}$

So, the two solutions are: $x = \frac{1 + 7}{4} = 2 \quad \text{and} \quad x = \frac{1 - 7}{4} = -\frac{3}{2}$

Step 3: Identify intervals and test

The solutions $x = 2$ and $x = -\frac{3}{2}$ divide the number line into three intervals:

• $(-\infty, -\frac{3}{2})$
• $(- \frac{3}{2}, 2)$
• $(2, \infty)$

Now, test a point from each interval in the inequality $2x^2 - x - 6 < 0$:

• For $x = -2$ (from $(-\infty, -\frac{3}{2})$): $2(-2)^2 - (-2) - 6 = 8 + 2 - 6 = 4 \quad (\text{not less than 0})$

• For $x = 0$ (from $(- \frac{3}{2}, 2)$): $2(0)^2 - (0) - 6 = -6 \quad (\text{less than 0})$

• For $x = 3$ (from $(2, \infty)$): $2(3)^2 - (3) - 6 = 18 - 3 - 6 = 9 \quad (\text{not less than 0})$

Step 4: Conclusion

The inequality holds for the interval $\left(-\frac{3}{2}, 2\right)$. Thus, the solution is: $\boxed{(-\frac{3}{2}, 2)}$

Would you like more details on any step or have any questions?

Here are 5 related questions to expand your understanding:

1. How does the quadratic formula work in solving inequalities?
2. What is the importance of testing intervals when solving inequalities?
3. How does the discriminant help determine the nature of the solutions?
4. Can this inequality be solved graphically, and how would that look?
5. How can you apply this method to solve other types of quadratic inequalities?

Tip: Always remember to test points from each interval when solving inequalities to check where the inequality holds true.