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Plz help this is hard!

 

Solve the inequality 4t^2 \le -9t + 12 - 4t + 23 + (2t - 1)(2t + 1). Write your answer in interval notation.

 Dec 18, 2023
 #1
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Let's solve the inequality step-by-step:

 

1. Expand the quadratic term:

 

4t^2 ≤ -9t + 12 - 4t + 23 + (4t^2 - 1)

 

2. Combine like terms:

 

8t^2 ≤ -13t + 34

 

3. Move all terms to one side:

 

8t^2 + 13t - 34 ≤ 0

 

4. Factor the inequality:

 

(8t - 7)(t + 4) ≤ 0

 

5. Analyze the factors and sign changes:

 

The inequality changes sign at the roots of the factors, which are t = 7/8 and t = -4. Additionally, the term (8t - 7) is positive for t > 7/8 and negative for t < 7/8. The term (t + 4) is positive for t > -4 and negative for t < -4.

 

6. Create a sign table:

 

Interval8t - 7t + 4(8t - 7)(t + 4)

t < -4--+

-4 < t < 7/8-+-

t > 7/8+++

 

 

7. Determine solution intervals:

 

The inequality is satisfied when (8t - 7)(t + 4) is non-positive. Based on the sign table:

 

The inequality is true for t ≤ -4 and t ≥ 7/8.

 

8. Write the solution in interval notation:

 

Therefore, the solution is the union of two intervals:

 

x ∈ (-∞, -4] ∪ [7/8, ∞)

 

This represents the set of all real numbers less than or equal to -4, and all real numbers greater than or equal to 7/8.

 Dec 18, 2023

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