The inequality is given as:

$5(x^2 - 1) > 24x$

Step 1: Expand the left side of the inequality

Distribute the $5$ across the terms inside the parentheses:

$5x^2 - 5 > 24x$

Step 2: Rearrange the inequality

Move all terms to one side so that the right side is $0$:

$5x^2 - 24x - 5 > 0$

Step 3: Solve the corresponding quadratic equation

To solve the inequality, first solve the quadratic equation:

$5x^2 - 24x - 5 = 0$

Use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 5$, $b = -24$, and $c = -5$. Substituting these values into the formula:

$x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4(5)(-5)}}{2(5)}$ $x = \frac{24 \pm \sqrt{576 + 100}}{10}$ $x = \frac{24 \pm \sqrt{676}}{10}$ $x = \frac{24 \pm 26}{10}$

Thus, the two solutions are:

$x_1 = \frac{24 + 26}{10} = \frac{50}{10} = 5$ $x_2 = \frac{24 - 26}{10} = \frac{-2}{10} = -0.2$

Step 4: Analyze the inequality

The quadratic inequality $5x^2 - 24x - 5 > 0$ has solutions when the quadratic expression is positive. The roots $x = 5$ and $x = -0.2$ divide the number line into three intervals:

1. $x < -0.2$
2. $-0.2 < x < 5$
3. $x > 5$

To determine where the inequality holds, test points from each interval in the expression $5x^2 - 24x - 5 > 0$:

• For $x = -1$ (in the interval $x < -0.2$):

$5(-1)^2 - 24(-1) - 5 = 5(1) + 24 - 5 = 24 > 0$ The inequality holds for $x < -0.2$.

• For $x = 0$ (in the interval $-0.2 < x < 5$):

$5(0)^2 - 24(0) - 5 = -5 < 0$ The inequality does not hold for $-0.2 < x < 5$.

• For $x = 6$ (in the interval $x > 5$):

$5(6)^2 - 24(6) - 5 = 180 - 144 - 5 = 31 > 0$ The inequality holds for $x > 5$.

Step 5: Final solution

The inequality $5x^2 - 24x - 5 > 0$ holds for:

$x < -0.2 \quad \text{or} \quad x > 5$

Thus, the solution to the inequality is:

$x \in (-\infty, -0.2) \cup (5, \infty)$

Let me know if you would like further details or explanations!

Related Questions:

1. How do we graph quadratic inequalities?
2. What is the quadratic formula and how does it work?
3. How do we factor quadratics if possible?
4. Can we solve inequalities graphically?
5. What is the significance of the discriminant in the quadratic formula?

Tip:

Always test points in different intervals to determine where a quadratic inequality holds true after solving the related equation.