We will solve the inequality:

$2x^2 - 9x - 5 \geq 0$

Step 1: Find the Roots of the Quadratic Equation

We solve the related equation:

$2x^2 - 9x - 5 = 0$

Using the quadratic formula:

$x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(2)(-5)}}{2(2)}$

$x = \frac{9 \pm \sqrt{81 + 40}}{4}$

$x = \frac{9 \pm \sqrt{121}}{4}$

$x = \frac{9 \pm 11}{4}$

Solving for both values:

$x = \frac{9 + 11}{4} = \frac{20}{4} = 5$

$x = \frac{9 - 11}{4} = \frac{-2}{4} = -\frac{1}{2}$

So, the roots are $x = 5$ and $x = -\frac{1}{2}$.

Step 2: Determine the Sign of the Quadratic Expression

The quadratic equation represents a parabola opening upwards (since the coefficient of $x^2$ is positive). The roots divide the number line into three intervals:

1. $(-\infty, -\frac{1}{2})$
2. $(-\frac{1}{2}, 5)$
3. $(5, \infty)$

We test a point from each interval:

• For $x = -1$ in $(-\infty, -\frac{1}{2})$:
  $2(-1)^2 - 9(-1) - 5 = 2(1) + 9 - 5 = 6 \quad \text{(positive)}$

• For $x = 0$ in $(-\frac{1}{2}, 5)$:
  $2(0)^2 - 9(0) - 5 = -5 \quad \text{(negative)}$

• For $x = 6$ in $(5, \infty)$:
  $2(6)^2 - 9(6) - 5 = 72 - 54 - 5 = 13 \quad \text{(positive)}$

Step 3: Write the Solution

We need the intervals where $2x^2 - 9x - 5 \geq 0$, meaning positive or zero values.
From the sign analysis, the quadratic expression is positive in the intervals $(-\infty, -\frac{1}{2}]$ and $[5, \infty)$ (including the roots since the inequality is "≥").

Thus, the final solution is:

$x \in (-\infty, -\frac{1}{2}] \cup [5, \infty)$

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Would you like a graphical explanation or more details?

Here are five related questions to explore:

1. How do we determine if a quadratic inequality opens upwards or downwards?
2. What is the difference between solving quadratic equations and quadratic inequalities?
3. How can we solve quadratic inequalities using a sign chart?
4. What is the role of the discriminant in solving quadratic inequalities?
5. How do we solve quadratic inequalities graphically?

Tip:

For inequalities, always check if you need to include or exclude the roots based on whether the inequality sign is strict ($<,>$) or inclusive ($\leq, \geq$).