Sure, let's graph the compound inequality [tex]\( x < 5 \text{ or } x \geq 8 \)[/tex].
1. Understand the inequality:
- The inequality [tex]\( x < 5 \)[/tex] means all numbers less than 5.
- The inequality [tex]\( x \geq 8 \)[/tex] means all numbers greater than or equal to 8.
2. Graph these on a number line:
- For [tex]\( x < 5 \)[/tex]:
- We draw an open circle at 5 (since 5 is not included).
- Shade all the region to the left of 5 (representing all numbers less than 5).
- For [tex]\( x \geq 8 \)[/tex]:
- We draw a closed circle at 8 (since 8 is included).
- Shade all the region to the right of 8 (representing all numbers greater than or equal to 8).
Here's how it looks on the number line:
[tex]\[ \text{-}\infty \, \, \, \, \, \, \, \mathbf{|}\, \, \, \, \, \, \mathbf{|}\, \, \, \, \, \, \mathbf{|}\mathbf{\circ}\qquad \, \, \mathbf{5}\, \, \, \, \, \, \mathbf{|} \, \, \, \, \, \, \mathbf{|}\, \, \, \, \, \, \mathbf{|} \, \mathbf{\bullet}\qquad \, \mathbf{8}\quad \mathbf{|}\mathbf{}}\text{:} \infty\][/tex]
- [tex]\(-\infty\)[/tex] to 5 (not including 5) is shaded.
- 8 to [tex]\(+\infty\)[/tex] (including 8) is shaded.
3. Verify the solution sets:
- For [tex]\( x < 5 \)[/tex], we include values like [tex]\(\{-10.0, -9.9, -9.8, \ldots, 4.9, 4.999999999999947\}\)[/tex].
- For [tex]\( x \geq 8 \)[/tex], we include values like [tex]\(\{8.099999999999937, 8.199999999999935, 8.299999999999933, \ldots, 14.8, 14.9\}\)[/tex].
4. Conclusion:
- The graph on the number line represents the solution where all values less than 5 and all values greater than or equal to 8 are included.
- This compound inequality does not include numbers between 5 and 8.
