Question
Solve the inequality. Write the solution set in interval notation if possible. \[ (4 d+5)^{2}>-6 \] The solution set is \( \square . \square \)
Ask by Watkins Hampton. in the United States
Mar 19,2025
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Tutor-Verified Answer
Answer
The solution set is all real numbers, written as \( (-\infty, \infty) \).
Solution
1. Recognize that for any real number \(d\), the expression \((4d+5)^2\) is a square, and thus it is always nonnegative:
\[
(4d+5)^2 \geq 0.
\]
2. Since \(0 > -6\), it follows that:
\[
(4d+5)^2 \geq 0 > -6.
\]
This means that the inequality \((4d+5)^2 > -6\) holds true for all real values of \(d\).
3. Therefore, the solution set in interval notation is:
\[
(-\infty, \infty).
\]
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Simplify this solution Extra Insights
Let's first consider the inequality \((4d + 5)^{2} > -6\). Noticing that the left side is a square, it is always non-negative, meaning it can never be less than zero. Thus, \((4d + 5)^{2} \geq 0\) for all real numbers \(d\). Since \(-6\) is less than \(0\), the inequality \((4d + 5)^{2} > -6\) is true for every real number \(d\). The solution set is \((-∞, ∞)\).
