To determine which inequality has an open circle when it is graphed on a number line, let's review each of the given inequalities and how they are represented on a number line.
1. [tex]\( x > \frac{3}{5} \)[/tex]
2. [tex]\( \frac{4}{7} \geq x \)[/tex]
3. [tex]\( x \leq 12 \)[/tex]
4. [tex]\( x \geq -6 \)[/tex]
An open circle is used in graphing inequalities to indicate that the number at that point is not included in the solution set. It is used for "greater than" (>) or "less than" (<) inequalities.
1. [tex]\( x > \frac{3}{5} \)[/tex]
- This inequality reads "x is greater than [tex]\(\frac{3}{5}\)[/tex]".
- Since it specifically states "greater than" and not "greater than or equal to", the number [tex]\(\frac{3}{5}\)[/tex] is not included in the solution.
- Therefore, when graphed on a number line, [tex]\(\frac{3}{5}\)[/tex] will have an open circle.
2. [tex]\( \frac{4}{7} \geq x \)[/tex]
- This inequality reads "[tex]\(\frac{4}{7}\)[/tex] is greater than or equal to x".
- The "greater than or equal to" sign ("≥") means that x can be equal to [tex]\(\frac{4}{7}\)[/tex], so this point will be included in the solution set.
- Therefore, this will have a closed circle on [tex]\(\frac{4}{7}\)[/tex].
3. [tex]\( x \leq 12 \)[/tex]
- This inequality reads "x is less than or equal to 12".
- The "less than or equal to" sign ("≤") means that x can be equal to 12, so this point will be included in the solution set.
- Therefore, this will have a closed circle on 12.
4. [tex]\( x \geq -6 \)[/tex]
- This inequality reads "x is greater than or equal to -6".
- The "greater than or equal to" sign ("≥") means that x can be equal to -6, so this point will be included in the solution set.
- Therefore, this will have a closed circle on -6.
After reviewing each inequality, we can see that the inequality with the open circle when graphed is:
[tex]\[ x > \frac{3}{5} \][/tex]
So, the answer is the first inequality, [tex]\( x > \frac{3}{5} \)[/tex].
