To solve the inequality \(-\frac{8}{m} \leq -8\), we can start by isolating \(m\).
Step 1: Multiply both sides by \(-1\) (remember to flip the inequality sign when multiplying by a negative number):
\[ \frac{8}{m} \geq 8 \]
Step 2: Next, we can rewrite the inequality:
\[ \frac{8}{m} \geq 8 \]
Step 3: Multiply both sides by \(m\) (note that we have to consider the sign of \(m\)):
- If \(m > 0\):
\[ 8 \geq 8m \ 1 \geq m \quad (dividing\ by\ 8) \]
- If \(m < 0\):
Flipping the direction of the inequality gives:
\[ 8m \geq 8 \ m \geq 1 \quad (dividing\ by\ 8) \]
In this case, since \(m < 0\), there are no solutions.
Combining both cases, the solution to the original inequality \( -\frac{8}{m} \leq -8 \) is:
\[ m \leq 1 \]
Plus, \(m\) must be positive to maintain the equality \(-\frac{8}{m} \frac{8}{m} > 0\).
Thus, the solution is:
\[ 0 < m \leq 1 \]
Examples of solutions are any values for \(m\) in the interval \((0, 1]\), such as \(0.5\), \(1\), etc.
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