To solve the inequality \(-2|x-7| \leq -8\), we can start by dividing both sides by \(-2\). Remember that when you divide an inequality by a negative number, you must reverse the inequality sign:
\[ |x - 7| \geq 4 \]
Now, we will solve for \(x\) using the definition of absolute value. The inequality \(|x - 7| \geq 4\) can be rewritten as two separate inequalities:
- \(x - 7 \geq 4\)
- \(x - 7 \leq -4\)
Solving the first inequality:
\[ x - 7 \geq 4 \] \[ x \geq 11 \]
Solving the second inequality:
\[ x - 7 \leq -4 \] \[ x \leq 3 \]
Now we can combine the two solutions. The solution set will be:
\[ x \leq 3 \quad \text{or} \quad x \geq 11 \]
In interval notation, this is:
\[ (-\infty, 3] \cup [11, \infty) \]
Thus, the solution to the inequality \(-2|x−7|≤−8\) is:
\[ (-\infty, 3] \cup [11, \infty) \]