Express the interval in​ set-builder notation and graph the interval on a number line.

Question

(6,9]

Select the correct choice below​ and, if​ necessasry, fill in the answer box to complete your choice.
a. The solution set in​ set-builder notation is {x blank}.
(Type an inequality or a compound​ inequality.)
b. The solution set is all real numbers.
c. There is no solution.

Answer 1

To express the interval \( (6, 9] \) in set-builder notation, we can write:

a. The solution set in set-builder notation is \( { x \mid 6 < x \leq 9 } \).

Explanation:

• The interval \( (6, 9] \) includes all real numbers \( x \) such that \( x \) is greater than 6 and less than or equal to 9.

Graphing the interval on a number line:

• To graph \( (6, 9] \):
  • Draw an open circle at 6 (indicating that 6 is not included).
  • Draw a closed circle at 9 (indicating that 9 is included).
  • Shade the region between these two points.

Here’s how it looks:

  <----|------|------|------|------|------|------>
       5      6      7      8      9      10
             o==================●

In this representation:

• \( o \) indicates that 6 is not included.
• \( ● \) indicates that 9 is included.