Question 1209291
**1. Rewrite the inequality**

* 4t² ≤ -9t + 12
* 4t² + 9t - 12 ≤ 0

**2. Find the roots of the quadratic equation**

* 4t² + 9t - 12 = 0
* (4t - 3)(t + 4) = 0 
* t = 3/4 or t = -4

**3. Determine the intervals**

* The roots divide the number line into three intervals:
    * Interval 1: t ≤ -4
    * Interval 2: -4 ≤ t ≤ 3/4
    * Interval 3: t ≥ 3/4

**4. Test points in each interval**

* **Interval 1 (t ≤ -4):** Choose t = -5. 
    * 4(-5)² + 9(-5) - 12 = 100 - 45 - 12 = 43 > 0 
* **Interval 2 (-4 ≤ t ≤ 3/4):** Choose t = 0.
    * 4(0)² + 9(0) - 12 = -12 ≤ 0 
* **Interval 3 (t ≥ 3/4):** Choose t = 1.
    * 4(1)² + 9(1) - 12 = 1 > 0

**5. Determine the solution**

* The inequality 4t² + 9t - 12 ≤ 0 is satisfied in **Interval 2**.

**6. Write the solution in interval notation**

* **Solution: [-4, 3/4]**

**Therefore, the solution to the inequality 4t² ≤ -9t + 12 in interval notation is [-4, 3/4].**