Question

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Part 1
Solve the inequality and express the solution set in​ set-builder notation and interval notation. Graph the solution set on the real number line.
4(2w−1​)≥6​(w+4)+2​(w−14​)

Answer 1

To solve the inequality \( 4(2w - 1) \geq 6(w + 4) + 2(w - 14) \), let’s start by simplifying both sides:

1. Distribute the terms:
\[
4(2w - 1) = 8w - 4
\]
\[
6(w + 4) = 6w + 24
\]
\[
2(w - 14) = 2w - 28
\]

2. Substitute these into the inequality:
\[
8w - 4 \geq 6w + 24 + 2w - 28
\]

3. Combine like terms on the right side:
\[
6w + 2w + 24 - 28 = 8w - 4
\]
So we can rewrite the inequality as:
\[
8w - 4 \geq 8w - 4
\]

4. Now, let's simplify the inequality:
Subtract \(8w\) from both sides:
\[
-4 \geq -4
\]
This statement is always true.

5. Thus, the solution includes all real numbers.

### Solution Set:
- In set-builder notation:
\[
\{w \,|\, w \in \mathbb{R}\}
\]

- In interval notation:
\[
(-\infty, \infty)
\]

### Graphing:
On the real number line, you would shade the entire line to indicate all real numbers are included in the solution set.

### Summary:
The solution to \( 4(2w - 1) \geq 6(w + 4) + 2(w - 14) \) is:
- Set-builder notation: \(\{ w \,|\, w \in \mathbb{R} \}\)
- Interval notation: \((- \infty, \infty)\)