Answer :
Let's analyze each inequality and determine which one will have an open circle when graphed on a number line.
1. [tex]\( x > \frac{3}{5} \)[/tex]:
- "Greater than" ([tex]\(>\)[/tex]) is a strict inequality, meaning that [tex]\( x \)[/tex] can be any value greater than [tex]\( \frac{3}{5} \)[/tex], but not equal to [tex]\( \frac{3}{5} \)[/tex].
- When graphed, a strict inequality is represented with an open circle at [tex]\( \frac{3}{5} \)[/tex], indicating that [tex]\( \frac{3}{5} \)[/tex] is not included in the solution set.
2. [tex]\( \frac{4}{7} \geq x \)[/tex]:
- "Greater than or equal to" ([tex]\(\geq\)[/tex]) is a non-strict inequality, meaning that [tex]\( x \)[/tex] can be equal to or less than [tex]\( \frac{4}{7} \)[/tex].
- When graphed, a non-strict inequality is represented with a closed circle at [tex]\( \frac{4}{7} \)[/tex], indicating that [tex]\( \frac{4}{7} \)[/tex] is included in the solution set.
3. [tex]\( x \leq 12 \)[/tex]:
- "Less than or equal to" ([tex]\(\leq\)[/tex]) is a non-strict inequality, meaning that [tex]\( x \)[/tex] can be equal to or less than [tex]\( 12 \)[/tex].
- When graphed, a non-strict inequality is represented with a closed circle at [tex]\( 12 \)[/tex], indicating that [tex]\( 12 \)[/tex] is included in the solution set.
4. [tex]\( x \geq -6 \)[/tex]:
- "Greater than or equal to" ([tex]\(\geq\)[/tex]) is a non-strict inequality, meaning that [tex]\( x \)[/tex] can be equal to or greater than [tex]\( -6 \)[/tex].
- When graphed, a non-strict inequality is represented with a closed circle at [tex]\( -6 \)[/tex], indicating that [tex]\( -6 \)[/tex] is included in the solution set.
The inequality [tex]\( x > \frac{3}{5} \)[/tex] involves a strict inequality and, therefore, will have an open circle when graphed on a number line.
Thus, the answer is:
[tex]\[ x > \frac{3}{5} \][/tex]
1. [tex]\( x > \frac{3}{5} \)[/tex]:
- "Greater than" ([tex]\(>\)[/tex]) is a strict inequality, meaning that [tex]\( x \)[/tex] can be any value greater than [tex]\( \frac{3}{5} \)[/tex], but not equal to [tex]\( \frac{3}{5} \)[/tex].
- When graphed, a strict inequality is represented with an open circle at [tex]\( \frac{3}{5} \)[/tex], indicating that [tex]\( \frac{3}{5} \)[/tex] is not included in the solution set.
2. [tex]\( \frac{4}{7} \geq x \)[/tex]:
- "Greater than or equal to" ([tex]\(\geq\)[/tex]) is a non-strict inequality, meaning that [tex]\( x \)[/tex] can be equal to or less than [tex]\( \frac{4}{7} \)[/tex].
- When graphed, a non-strict inequality is represented with a closed circle at [tex]\( \frac{4}{7} \)[/tex], indicating that [tex]\( \frac{4}{7} \)[/tex] is included in the solution set.
3. [tex]\( x \leq 12 \)[/tex]:
- "Less than or equal to" ([tex]\(\leq\)[/tex]) is a non-strict inequality, meaning that [tex]\( x \)[/tex] can be equal to or less than [tex]\( 12 \)[/tex].
- When graphed, a non-strict inequality is represented with a closed circle at [tex]\( 12 \)[/tex], indicating that [tex]\( 12 \)[/tex] is included in the solution set.
4. [tex]\( x \geq -6 \)[/tex]:
- "Greater than or equal to" ([tex]\(\geq\)[/tex]) is a non-strict inequality, meaning that [tex]\( x \)[/tex] can be equal to or greater than [tex]\( -6 \)[/tex].
- When graphed, a non-strict inequality is represented with a closed circle at [tex]\( -6 \)[/tex], indicating that [tex]\( -6 \)[/tex] is included in the solution set.
The inequality [tex]\( x > \frac{3}{5} \)[/tex] involves a strict inequality and, therefore, will have an open circle when graphed on a number line.
Thus, the answer is:
[tex]\[ x > \frac{3}{5} \][/tex]
