Question

reasoning how are the graphs of ( k < 0 ) and ( k leq 0 ) alike? how are they different?

• both have a circle at 0 and an arrow to the left; ( k < 0 ) has an open circle, ( k leq 0 ) has a closed circle
• both have a circle at 0 and an arrow to the right; ( k < 0 ) has an open circle, ( k leq 0 ) has a closed circle
• both have a circle at 0 and an arrow to the right; ( k < 0 ) has an closed circle, ( k leq 0 ) has a open circle
• both have a circle at 0 and an arrow to the left; ( k < 0 ) has an closed circle, ( k leq 0 ) has a open circle

Explanation:

1. Analyze the inequality \( k < 0 \): This represents all real numbers less than 0. On a number line, we use an open circle at 0 (since 0 is not included) and an arrow pointing to the left (towards negative numbers).
2. Analyze the inequality \( k \leq 0 \): This represents all real numbers less than or equal to 0. On a number line, we use a closed circle at 0 (since 0 is included) and an arrow pointing to the left (towards negative numbers).
3. Compare the two: Both have a circle at 0 and an arrow to the left. The difference is that \( k < 0 \) has an open circle (excludes 0) and \( k \leq 0 \) has a closed circle (includes 0). So the first option is correct.

Answer:

A. Both have a circle at 0 and an arrow to the left; \( k < 0 \) has an open circle, \( k \leq 0 \) has a closed circle