Identify solution(s) to the system below.

y≤−12x2−7x−2
y≤2x−8

Select all that apply.

Question

Identify solution(s) to the system below.

y≤−12x2−7x−2
y≤2x−8

Select all that apply.

BlackTom AI · Accepted Answer

We need points (x,y) that satisfy both inequalities: y ≤ -12x^2 - 7x - 2 and y ≤ 2x - 8. For each option, check whether the y-value at that x is less than or equal to both bounds by taking the lesser of the two right-hand sides for that x.
Option A: (-1, -9)
- Compute the first bound at x = -1: -12(-1)^2 - 7(-1) - 2 = -12 + 7 - 2 = -7.
- Compute the second bound at x = -1: 2(-1) - 8 = -2 - 8 = -10.
- The system requires y ≤ min(-7, -10) = -10. Since -9 is not ≤ -10, this point fails the second inequality as well as the combined bound. Thus this option is not a solution.

Option B: (-1, -11)
- At x = -1, the bounds are -7 (from the quadratic) and -10 (from the line). The combined requirement is y ≤ -10.
- Here y = -11, which satisfies y ≤ -10. Therefore this point satisfies both inequalities and is a valid solution.

Option C: (-8, 0)
- At x = -8, compute the quadratic bound: -12(64) - 7(-8) - 2 = -768 + 56 - 2 = -714.
- The linear bound: 2(-8) - 8 = -16 - 8 = -24.
- The combined requirement is y ≤ min(-714, -24) = -714.
- With y = 0, we have 0 ≤ -714, which is false. So this point does not satisfy the system.

Option D: (0, -8)
- At x = 0, the quadratic bound is -12(0)^2 - 7(0) - 2 = -2.
- The linear bound is 2(0) - 8 = -8.
- The combined requirement is y ≤ min(-2, -8) = -8.
- Here y = -8, and -8 ≤ -8 is true, so this point satisfies both inequalities and is a valid solution.

Summary of checks:
- A: (-1, -9) fails the necessary bound (not ≤ -10).
- B: (-1, -11) satisfies y ≤ -10, so valid.
- C: (-8, 0) fails since 0 is not ≤ -714.
- D: (0, -8) satisfies y ≤ -8, so valid.

Therefore, the solution set includes the points (-1, -11) and (0, -8).

Identify solution(s) to the system below. y≤−12x2−7x−2 y≤2x−8 Select all that apply. 多项选择题

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