Certainly! Let's carefully examine the inequality given and explore alternative ways to express it.
The inequality provided is:
[tex]\[ -12 < x < 7 \][/tex]
This inequality describes the set of all real numbers [tex]\( x \)[/tex] that lie strictly between -12 and 7. Now, let's consider the different ways we can represent this solution:
1. Interval Notation:
The interval notation for this inequality is:
[tex]\[ (-12, 7) \][/tex]
This notation clearly indicates that [tex]\( x \)[/tex] takes values greater than -12 and less than 7.
2. Set-Builder Notation:
Set-builder notation is another way to express this inequality, where we describe the set of numbers [tex]\( x \)[/tex] that satisfy a certain condition:
[tex]\[ \{ x \mid -12 < x < 7 \} \][/tex]
This reads as "the set of all [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than -12 and less than 7."
3. Union of Intervals:
The union of intervals is not appropriate here because the solution is a single interval:
[tex]\[ (-\infty, -12) \cup (7, \infty) \][/tex]
This represents values outside the interval from -12 to 7, which does not correspond to our given inequality.
4. Incorrect Set-Builder Form:
The expression below misrepresents the inequality because it combines two separate conditions with "or":
[tex]\[ \{ x \mid x > -12 \text{ or } x < 7 \} \][/tex]
This statement includes values that are less than -12 or greater than 7, which also does not meet our specified range.
Given these representations, the correct alternative way to show Fuaad's solution is:
[tex]\[ \{ x \mid -12 < x < 7 \} \][/tex]
So, the answer is:
[tex]\[ \{ x \mid -12 < x < 7 \} \][/tex]
This representation succinctly and accurately describes the set of values for [tex]\( x \)[/tex] that satisfy the given inequality.
