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Bot GPT-4 Turbo To solve the inequality \(16 ≥ -5x - 9 ≥ -44\), we need to solve for \(x\) in two parts: one for \(16 ≥ -5x - 9\) and the other for \(-5x - 9 ≥ -44\).
Let's begin with the first part of the inequality:
1. \(16 ≥ -5x - 9\)
Add \(9\) to each side of the inequality to isolate the term with \(x\) on one side:
\[16 + 9 ≥ -5x\]
\[25 ≥ -5x\]
Now, divide by \(-5\) to solve for \(x\). Note that dividing by a negative number will flip the inequality sign:
\[\frac{25}{-5} ≤ x\]
\[-5 ≤ x\]
\[x ≥ -5\] (After flipping the inequality)
Now let's solve the second part of the inequality:
2. \(-5x - 9 ≥ -44\)
Add \(9\) to each side of the inequality:
\[-5x ≥ -44 + 9\]
\[-5x ≥ -35\]
Now, divide by \(-5\), remembering to flip the inequality sign again:
\[\frac{-35}{-5} ≤ x\]
\[7 ≤ x\]
\[x ≥ 7\]
Now we combine both parts of the inequality to find the values of \(x\) that satisfy both conditions:
\[x ≥ -5 \text{ and } x ≥ 7\]
The larger inequality condition \(x ≥ 7\) is the more restrictive one, so this is our final solution.
To graph this solution on a number line:
---|----------|----------|---->
-5 0 7
We draw a solid circle or a closed dot on \(7\) (since \(7\) is included in the solution due to the "greater than or equal to" sign) and shade the number line to the right of \(7\) to indicate all the values of \(x\) that are greater than or equal to \(7\).
Here's a visual representation (although not to scale):
```plaintext
<----------------|)
-5 0 1 2 3 4 5 6 7 8 9 10 ...
```
The parentheses ")" at the left end indicate that the number line continues indefinitely to the left, and the line is not shaded from -5 to 7 because those numbers are not included in the solution set. The shaded region starts at 7 (inclusive) and extends to the right indefinitely.