SOLUTION: Solve the inequality 4t^2 \le -9t + 12. Write your answer in interval notation.

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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].**
Answer by ikleyn(52817)   (Show Source): You can put this solution on YOUR website!
.
Solve the inequality 4t^2 \le -9t + 12. Write your answer in interval notation.
~~~~~~~~~~~~~~~~~~~~~~~


        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

Solved.



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