Answer:
Option 4: (0, 5]
Step-by-step explanation:
Given the <u>inequality notation</u>, 0 < <em>x</em> ≤ 5 :
<h2><u>
Definitions:</u></h2><h3>
open- and closed-parenthesis "
( )"</h3>
We know that using the <u>open- and closed-parenthesis</u>, "( )" are used to express interval notation for which the endpoints are <em><u>not</u></em> included as part of the solution. Hence, the open- and closed-parenthesis are associated with the inequality symbols, <em>less than </em>(>) or greater than (<), as you do not include the endpoints when these inequality symbols are used.
<h3>
open- and closed-brackets "
[ ]"</h3>
In contrast, the <u>open- and closed-brackets</u>, "[ ]" are used to express interval notation for which the endpoints <u><em>are</em></u> included as part of the solution. Hence, the open- and closed-parenthesis are associated with the inequality symbols, <em>less than or equal to </em>(≤), and <em>greater than</em><em> </em><em>or equal to</em> (≥).
<h2><u>Explanation:</u></h2>
In the inequality notation, 0 < <em>x</em> ≤ 5, the endpoints are 0 and 5.
The inequality symbol <u>between 0 and x</u> is "<," which means that we must use an open parenthesis, " ( " for the endpoint, 0.
While the inequality symbol <u>between x and 5</u> is "≤," which means that we must use an closed bracket, " ] " for the endpoint, 5.
Therefore, the correct answer is Option 4: (0, 5].
