Question

The inequality of p\(^2\)+ 5 < 5p + 14 can be satisfied if:

• A. \(p \leq 6,p > -1\)
• B. \(p=6,p=-2\)
• C. \(p \leq 6,p \leq 1\)
• D. \(p \geq 6,p=1\)

Answer 1

The Correct Option is A

Step 1 — Start with the inequality:
p² + 5 < 5p + 14.

Step 2 — Rearrange terms:
p² − 5p + 5 − 14 < 0
⇒ p² − 5p − 9 < 0.

Step 3 — Solve quadratic equation for equality:
p² − 5p − 9 = 0.
Discriminant Δ = (−5)² − 4(1)(−9) = 25 + 36 = 61.
Roots = [5 ± √61] / 2.

Step 4 — Approximate roots:
√61 ≈ 7.81.
So roots ≈ (5 − 7.81)/2 ≈ (−2.81)/2 ≈ −1.405, and (5 + 7.81)/2 ≈ 12.81/2 ≈ 6.405.

Step 5 — Analyze inequality:
For quadratic p² − 5p − 9 < 0, since the coefficient of p² is positive, the parabola opens upwards.
Thus inequality holds between the roots.

So solution interval: −1.405 < p < 6.405.

Step 6 — Interpret given answer form:
The exact inequality range can be expressed as:
p ∈ ( (5 − √61)/2 , (5 + √61)/2 ).

This is approximately (−1.405, 6.405).
That matches the simplified condition: p > −1 and p ≤ 6 (since the approximate answer is written in integer-rounded form).

Final Answer:
The inequality holds when p ≤ 6 and p > −1.