Question

solve the compound inequality. graph the two inequalities on the first two number lines and the solution set on the third number line. x < 5 or x < 6 graph the inequality x < 5. choose the correct graph below.

Explanation:

Step1: Analyze \( x < 5 \)

For \( x < 5 \), the graph has an open circle at \( 5 \) (since \( 5 \) is not included) and an arrow pointing left (for values less than \( 5 \)).

Step2: Analyze \( x < 6 \)

For \( x < 6 \), the graph has an open circle at \( 6 \) (not included) and an arrow pointing left. But since we have "or", we combine the two graphs. The solution set for \( x < 5 \) or \( x < 6 \) is just \( x < 6 \)? Wait, no—wait, \( x < 5 \) is a subset of \( x < 6 \). Wait, no, the compound inequality is \( x < 5 \) or \( x < 6 \). The "or" means the union. Since all \( x < 5 \) are also \( x < 6 \), the solution is \( x < 6 \)? Wait, no, maybe I misread. Wait, the first part is graphing \( x < 5 \) and \( x < 6 \) on first two, then solution on third. Wait, the question about graphing \( x < 5 \): the correct graph for \( x < 5 \) has an open circle at \( 5 \) and arrow left. Looking at the options: Option B? Wait, no, let's check the graphs. Option B: open circle at 5, arrow left (and also right? No, wait the first two graphs for \( x < 5 \): let's see the options. Wait, the problem says "Graph the inequality \( x < 5 \). Choose the correct graph below." The inequality \( x < 5 \) has an open circle at \( 5 \) (because \( 5 \) is not included) and the arrow points to the left (for values less than \( 5 \)). Looking at the options:

- Option A: closed circle at 5 (wrong, since \( x < 5 \) is open)
- Option B: open circle at 5, arrow left (and also the line is from -10 to 5 open, then arrow left? Wait, no, the graph for \( x < 5 \) should have an open circle at 5 and all numbers less than 5 shaded. So the correct graph for \( x < 5 \) is the one with open circle at 5 and arrow left (so among the options, let's see: the first row, second column? Wait, the options are A, B, C, D. Let's re-express:

- A: closed circle at 5, arrow left (includes 5, wrong)
- B: open circle at 5, arrow left (and the line is shaded left of 5, open circle)
- C: closed circle at 5, arrow right (wrong)
- D: open circle at 5, arrow right (wrong)

So the correct graph for \( x < 5 \) is Option B.

Then, for the compound inequality \( x < 5 \) or \( x < 6 \): the solution set is all real numbers (since \( x < 5 \) is a subset of \( x < 6 \), so "or" means union, which is \( x < 6 \)). But maybe the problem is first graph \( x < 5 \) (open circle 5, left), then graph \( x < 6 \) (open circle 6, left), then the solution set (union) is \( x < 6 \), but maybe the initial question is about graphing \( x < 5 \) first.

Wait, the user's question is to solve the compound inequality \( x < 5 \) or \( x < 6 \), graph the two inequalities on first two number lines, and solution on third. But also, there's a sub-question: "Graph the inequality \( x < 5 \). Choose the correct graph below."

For \( x < 5 \):

- Open circle at 5 (since \( 5 \) is not a solution)
- Shade all numbers less than 5 (arrow to the left)

Looking at the options:

- A: Closed circle at 5 (incorrect, as \( x < 5 \) does not include 5)
- B: Open circle at 5, shaded left (correct)
- C: Closed circle at 5, shaded right (incorrect)
- D: Open circle at 5, shaded right (incorrect)

So the correct graph for \( x < 5 \) is Option B.

For the compound inequality \( x < 5 \) or \( x < 6 \):

- The solution to \( x < 5 \) is all \( x < 5 \)
- The solution to \( x < 6 \) is all \( x < 6 \)
- The "or" means the union of these two sets. Since all \( x < 5 \) are already included in \( x < 6 \), the union is \( x < 6 \)
- To graph \( x < 6 \): open circle at 6, shaded left (all numbers less than 6)

But the initial problem's first part is to graph \( x < 5 \) (open circle 5, left) and \( x < 6 \) (open circle 6, left) on first two, then solution (open circle 6, left) on third.

But the specific question here is "Graph the inequality \( x < 5 \). Choose the correct graph below." So the answer is the graph with open circle at 5 and arrow left, which is Option B.

Answer:

For graphing \( x < 5 \), the correct option is B (open circle at 5, arrow pointing left, shading all values less than 5). For the compound inequality \( x < 5 \) or \( x < 6 \), the solution set is \( x < 6 \) (graph with open circle at 6, arrow left).