Question

Fill the blank with either \( E \) or \( \& \) to make the statement true. \[ -7 \quad\{12-5,12\} \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Answer 1

The correct fill for the blank is
`\( -7 \quad \{12-5 \, \& \, 12\} \)`, indicating that -7 is within the inclusive range of real numbers from 5 to 12.

Step-by-step explanation:

In this set notation, we are looking for a number that is less than or equal to 12 and greater than or equal to 5. The interval notation
`\([5,12]\)` includes all real numbers between 5 and 12, including the endpoints. The curly braces
`\(\{ \}\)` indicate a set, and since we need a number within the given interval, we should use the "and" operator, which is represented by the symbol
`\( \& \)`. Therefore, the correct fill for the blank is:

`\[ -7 \quad \{12-5 \, \& \, 12\} \]`

This notation signifies that -7 is within the range from 5 to 12, inclusive.

Now, let's explain further. The set notation
`\(\{12-5,12\}\)` is equivalent to
`\(\{-7,12\}\)`, representing the set of numbers -7 and 12. The interval notation
`\([5,12]\)` includes all real numbers from 5 to 12. When we combine these notations with the "and" operator & , we ensure that we are considering the intersection of the two sets. In this case, the intersection is
`\{-7,12\}`, indicating that -7 is indeed within the specified range. Therefore, the expression
`\(-7 \quad \{12-5 \, \& \, 12\}\)` accurately conveys that -7 is part of the interval
`\([5,12]\)`.