To determine which inequality has an open circle when it is graphed on a number line, let's analyze each inequality and identify whether it uses an open or a closed circle.
1. [tex]\( x > \frac{3}{5} \)[/tex]:
- The inequality [tex]\( x > \frac{3}{5} \)[/tex] tells us that [tex]\( x \)[/tex] is greater than [tex]\( \frac{3}{5} \)[/tex].
- When graphed on a number line, an open circle is used at [tex]\( \frac{3}{5} \)[/tex] to indicate that [tex]\( \frac{3}{5} \)[/tex] is not included in the solution set.
2. [tex]\( \frac{4}{7} \geq x \)[/tex]:
- The inequality [tex]\( \frac{4}{7} \geq x \)[/tex] tells us that [tex]\( \frac{4}{7} \)[/tex] is greater than or equal to [tex]\( x \)[/tex].
- When graphed on a number line, a closed circle is used at [tex]\( \frac{4}{7} \)[/tex] to indicate that [tex]\( \frac{4}{7} \)[/tex] is included in the solution set.
3. [tex]\( x \leq 12 \)[/tex]:
- The inequality [tex]\( x \leq 12 \)[/tex] tells us that [tex]\( x \)[/tex] is less than or equal to 12.
- When graphed on a number line, a closed circle is used at 12 to indicate that 12 is included in the solution set.
4. [tex]\( x \geq -6 \)[/tex]:
- The inequality [tex]\( x \geq -6 \)[/tex] tells us that [tex]\( x \)[/tex] is greater than or equal to -6.
- When graphed on a number line, a closed circle is used at -6 to indicate that -6 is included in the solution set.
Based on this analysis, the inequality that uses an open circle when it is graphed on a number line is:
[tex]\[ x > \frac{3}{5} \][/tex]
The first inequality, [tex]\( x > \frac{3}{5} \)[/tex], has an open circle.
