Question
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Tutor-Verified Answer
Answer
\( k \geq 0 \) or \( k < -8 \).
Solution
Solve the equation by following steps:
- step0: Solve for \(k\):
\(9k=0\)
- step1: Rewrite the expression:
\(k=0\)
Solve the equation \( k+5=-3 \).
Solve the equation by following steps:
- step0: Solve for \(k\):
\(k+5=-3\)
- step1: Move the constant to the right side:
\(k=-3-5\)
- step2: Subtract the numbers:
\(k=-8\)
Let's solve the inequalities step by step.
1. **First Inequality: \( 9k \geq 0 \)**
To solve this inequality, we can set it equal to zero to find the boundary point:
\[
9k = 0 \implies k = 0
\]
Since \( 9k \) is greater than or equal to zero, \( k \) can be any value greater than or equal to 0. Thus, the solution for this inequality is:
\[
k \geq 0
\]
2. **Second Inequality: \( k + 5 < -3 \)**
We can solve this by isolating \( k \):
\[
k + 5 = -3 \implies k = -3 - 5 \implies k = -8
\]
Since this is a strict inequality, the solution is:
\[
k < -8
\]
### Final Solution
The overall solution to the compound inequality \( 9k \geq 0 \) or \( k + 5 < -3 \) is:
- \( k \geq 0 \) or \( k < -8 \)
This means \( k \) can take any value in the ranges \( (-\infty, -8) \) or \( [0, \infty) \).
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Simplify this solution The Deep Dive
To solve this compound inequality, let's take each part one at a time! The first part, \( 9k \geq 0 \), tells us that \( k \) can be zero or positive, meaning \( k \geq 0 \). Now, for the second part, \( k + 5 < -3 \), we can isolate \( k \) by subtracting 5 from both sides, giving us \( k < -8 \). So, combining these results, the solution is that \( k \) is either any non-negative number (from the first inequality) or any number less than -8 (from the second inequality). You could represent this in interval notation as \( (-\infty, -8) \cup [0, \infty) \).
