Certainly! To find the domain of the expression [tex]$\frac{37}{2y + 7}$[/tex], we need to determine the values of [tex]\( y \)[/tex] for which the expression is defined. The key issue here is the denominator, since division by zero is undefined.
So, we have to ensure that the denominator [tex]\( 2y + 7 \neq 0 \)[/tex].
Let's find when the denominator equals zero:
[tex]\[
2y + 7 = 0
\][/tex]
To solve for [tex]\( y \)[/tex], follow these steps:
1. Subtract 7 from both sides of the equation:
[tex]\[
2y + 7 - 7 = 0 - 7
\][/tex]
[tex]\[
2y = -7
\][/tex]
2. Divide both sides by 2:
[tex]\[
y = \frac{-7}{2}
\][/tex]
So, the expression [tex]\( \frac{37}{2y + 7} \)[/tex] is undefined when [tex]\( y = -\frac{7}{2} \)[/tex]. For all other values of [tex]\( y \)[/tex], the expression is defined.
Therefore, the domain of the expression [tex]\( \frac{37}{2y + 7} \)[/tex] is all real numbers except [tex]\( y = -\frac{7}{2} \)[/tex].
In interval notation, the domain can be written as:
[tex]\[
(-\infty, -\frac{7}{2}) \cup (-\frac{7}{2}, \infty)
\][/tex]
Thus, the domain of the expression is [tex]\( y \neq -\frac{7}{2} \)[/tex].
