Question 1209291: Solve the inequality 4t^2 \le -9t + 12. Write your answer in interval notation. Found 2 solutions by textot, ikleyn:Answer by textot(100) (Show Source):
You can put this solution on YOUR website! **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].**
You can put this solution on YOUR website! .
Solve the inequality 4t^2 \le -9t + 12. Write your answer in interval notation.
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The solution by the other tutor in his post is incorrect.
The mistake is in incorrect factoring of the equation (which, actually, can not be factored).
I came to bring a correct solution.
Write in the standard form quadratic inequality
4t^2 + 9t - 12 <= 0. (1)
The corresponding quadratic equation is
4t^2 + 9t - 12 = 0.
Find its roots using the quadratic formula
= = .
The roots are and .
The quadratic function in the left side of (1) is non-negative between the roots
<= t <=
In the interval form, the solution set is the union of two sets
(,] U [,). ANSWER
(
