To determine which inequality has an open circle when it is graphed on a number line, let's consider the concept of open and closed circles on number lines:
1. An open circle is used to represent that the value at that point is not included in the solution. This typically occurs with the inequalities using [tex]\(<\)[/tex] or [tex]\(>\)[/tex] symbols.
2. A closed circle is used to denote that the value at that point is included in the solution. This typically occurs with the inequalities using [tex]\(\leq\)[/tex] or [tex]\(\geq\)[/tex] symbols.
Let's analyze each inequality provided:
1. [tex]\(x > \frac{3}{5}\)[/tex]: This inequality means that [tex]\(x\)[/tex] should be greater than [tex]\(\frac{3}{5}\)[/tex] but not equal to [tex]\(\frac{3}{5}\)[/tex]. Therefore, an open circle will be used at [tex]\(\frac{3}{5}\)[/tex] because [tex]\(\frac{3}{5}\)[/tex] itself is not included in the solution set.
2. [tex]\(\frac{4}{7} \geq x\)[/tex]: This inequality means that [tex]\(x\)[/tex] should be less than or equal to [tex]\(\frac{4}{7}\)[/tex]. Therefore, a closed circle will be used at [tex]\(\frac{4}{7}\)[/tex] because [tex]\(\frac{4}{7}\)[/tex] is included in the solution set.
3. [tex]\(x \leq 12\)[/tex]: This inequality means that [tex]\(x\)[/tex] should be less than or equal to 12. Therefore, a closed circle will be used at 12 because 12 is included in the solution set.
4. [tex]\(x \geq -6\)[/tex]: This inequality means that [tex]\(x\)[/tex] should be greater than or equal to -6. Therefore, a closed circle will be used at -6 because -6 is included in the solution set.
Given the analysis above, the inequality that has an open circle when graphed on a number line is:
[tex]\[ x > \frac{3}{5} \][/tex]
