Answer :
To determine which inequality has an open circle when it is graphed on a number line, we first need to understand what an open circle represents in the context of inequalities.
An open circle is used on a number line to indicate that a particular value is not included in the set of solutions for the inequality. This occurs with strict inequalities, which use the symbols '<' (less than) or '>' (greater than). In other words:
- If an inequality uses '<' or '>', it will have an open circle at the boundary value.
- If an inequality uses '≤' (less than or equal to) or '≥' (greater than or equal to), it will have a closed circle at the boundary value, indicating that the boundary value is included in the set of solutions.
Now, let’s examine each of the given inequalities:
1. [tex]\( x > \frac{3}{5} \)[/tex]:
- This inequality states that [tex]\( x \)[/tex] is greater than [tex]\(\frac{3}{5}\)[/tex].
- The boundary value here is [tex]\(\frac{3}{5}\)[/tex], and since the inequality uses '>', the value [tex]\( \frac{3}{5} \)[/tex] is not included in the solution set.
- Therefore, this inequality will have an open circle at [tex]\(\frac{3}{5}\)[/tex].
2. [tex]\( \frac{4}{7} \geq x \)[/tex]:
- This inequality states that [tex]\(\frac{4}{7} \)[/tex] is greater than or equal to [tex]\( x \)[/tex].
- The boundary value here is [tex]\(\frac{4}{7}\)[/tex], and since the inequality uses '≥', the value [tex]\(\frac{4}{7}\)[/tex] is included in the solution set.
- Therefore, this inequality will have a closed circle at [tex]\(\frac{4}{7}\)[/tex].
3. [tex]\( x \leq 12 \)[/tex]:
- This inequality states that [tex]\( x \)[/tex] is less than or equal to [tex]\( 12 \)[/tex].
- The boundary value here is [tex]\( 12 \)[/tex], and since the inequality uses '≤', the value [tex]\( 12 \)[/tex] is included in the solution set.
- Therefore, this inequality will have a closed circle at [tex]\( 12 \)[/tex].
4. [tex]\( x \geq -6 \)[/tex]:
- This inequality states that [tex]\( x \)[/tex] is greater than or equal to [tex]\(-6\)[/tex].
- The boundary value here is [tex]\(-6\)[/tex], and since the inequality uses '≥', the value [tex]\(-6\)[/tex] is included in the solution set.
- Therefore, this inequality will have a closed circle at [tex]\(-6\)[/tex].
Based on this analysis, the inequality that has an open circle when it is graphed on a number line is:
[tex]\[ x > \frac{3}{5} \][/tex]
An open circle is used on a number line to indicate that a particular value is not included in the set of solutions for the inequality. This occurs with strict inequalities, which use the symbols '<' (less than) or '>' (greater than). In other words:
- If an inequality uses '<' or '>', it will have an open circle at the boundary value.
- If an inequality uses '≤' (less than or equal to) or '≥' (greater than or equal to), it will have a closed circle at the boundary value, indicating that the boundary value is included in the set of solutions.
Now, let’s examine each of the given inequalities:
1. [tex]\( x > \frac{3}{5} \)[/tex]:
- This inequality states that [tex]\( x \)[/tex] is greater than [tex]\(\frac{3}{5}\)[/tex].
- The boundary value here is [tex]\(\frac{3}{5}\)[/tex], and since the inequality uses '>', the value [tex]\( \frac{3}{5} \)[/tex] is not included in the solution set.
- Therefore, this inequality will have an open circle at [tex]\(\frac{3}{5}\)[/tex].
2. [tex]\( \frac{4}{7} \geq x \)[/tex]:
- This inequality states that [tex]\(\frac{4}{7} \)[/tex] is greater than or equal to [tex]\( x \)[/tex].
- The boundary value here is [tex]\(\frac{4}{7}\)[/tex], and since the inequality uses '≥', the value [tex]\(\frac{4}{7}\)[/tex] is included in the solution set.
- Therefore, this inequality will have a closed circle at [tex]\(\frac{4}{7}\)[/tex].
3. [tex]\( x \leq 12 \)[/tex]:
- This inequality states that [tex]\( x \)[/tex] is less than or equal to [tex]\( 12 \)[/tex].
- The boundary value here is [tex]\( 12 \)[/tex], and since the inequality uses '≤', the value [tex]\( 12 \)[/tex] is included in the solution set.
- Therefore, this inequality will have a closed circle at [tex]\( 12 \)[/tex].
4. [tex]\( x \geq -6 \)[/tex]:
- This inequality states that [tex]\( x \)[/tex] is greater than or equal to [tex]\(-6\)[/tex].
- The boundary value here is [tex]\(-6\)[/tex], and since the inequality uses '≥', the value [tex]\(-6\)[/tex] is included in the solution set.
- Therefore, this inequality will have a closed circle at [tex]\(-6\)[/tex].
Based on this analysis, the inequality that has an open circle when it is graphed on a number line is:
[tex]\[ x > \frac{3}{5} \][/tex]
