Let's solve the inequality:

$2000 - t^2 - 140t > 500$

Step 1: Move all terms to one side of the inequality.

First, subtract 500 from both sides:

$2000 - t^2 - 140t - 500 > 0$

Simplify:

$1500 - t^2 - 140t > 0$

Step 2: Rearrange the terms.

$-t^2 - 140t + 1500 > 0$

To make the calculation simpler, multiply the entire inequality by $-1$ (this reverses the inequality sign):

$t^2 + 140t - 1500 < 0$

Step 3: Solve the quadratic equation.

We'll solve the related quadratic equation $t^2 + 140t - 1500 = 0$ using the quadratic formula:

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

For the equation $t^2 + 140t - 1500 = 0$, we have:

• $a = 1$
• $b = 140$
• $c = -1500$

Substitute these values into the quadratic formula:

$t = \frac{-140 \pm \sqrt{140^2 - 4(1)(-1500)}}{2(1)}$ $t = \frac{-140 \pm \sqrt{19600 + 6000}}{2}$ $t = \frac{-140 \pm \sqrt{25600}}{2}$ $t = \frac{-140 \pm 160}{2}$

Now, solve for both values of $t$:

1. $t = \frac{-140 + 160}{2} = \frac{20}{2} = 10$
2. $t = \frac{-140 - 160}{2} = \frac{-300}{2} = -150$

So the solutions to the quadratic equation are $t = 10$ and $t = -150$.

Step 4: Determine the solution to the inequality.

We now have the two critical points $t = 10$ and $t = -150$. To solve $t^2 + 140t - 1500 < 0$, we check the sign of the expression in the three intervals created by these points:

1. For $t < -150$, choose $t = -200$: $(-200)^2 + 140(-200) - 1500 = 40000 - 28000 - 1500 = 10500 > 0$ This is positive, so the expression is positive in this interval.

2. For $-150 < t < 10$, choose $t = 0$: $(0)^2 + 140(0) - 1500 = -1500 < 0$ This is negative, so the expression is negative in this interval.

3. For $t > 10$, choose $t = 20$: $(20)^2 + 140(20) - 1500 = 400 + 2800 - 1500 = 1700 > 0$ This is positive, so the expression is positive in this interval.

Step 5: Write the solution.

The quadratic expression is less than 0 in the interval $-150 < t < 10$. Therefore, the solution to the inequality is:

$-150 < t < 10$

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

Follow-up Questions:

1. How do you solve a quadratic inequality graphically?
2. What is the significance of critical points in quadratic inequalities?
3. Can you solve quadratic inequalities using factoring instead of the quadratic formula?
4. What happens if the inequality is non-strict (e.g., $\leq$ or $\geq$)?
5. How do the properties of quadratic functions influence the solution to inequalities?

Tip: Always remember to flip the inequality sign when multiplying or dividing by a negative number in inequalities!