This interval indicates that
can be any value greater than 6.48, extending to infinity.
To solve the given equation for
, where
and the equation is:
![\[ |y| \cdot |y| = 42 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/ku5qrgox0xwzyxjh0obom26j0aix8crhd0.png)
We will work through the solution step-by-step. Given that
, the absolute value of
, which is
, will simply be
. This simplifies our equation to:
![\[ y \cdot y = 42 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/xha1x7wisnyniha2606rtmezjh29qyjh8k.png)
or
![\[ y^2 = 42 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/534rn3nboz6y1aaj7ppyb2opgs1h2cnlyv.png)
Now, let's solve for
. Since we are only considering positive values for
, we only need the positive square root of 42. Let's calculate that.
The positive value of
that satisfies the equation
is approximately 6.48. Since we are asked for the range in interval notation and considering only positive values of
, the range would be:
![\[ (0, \infty) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/8p8p1nbxidmgs0q2cob9c1zadnxa49g4jx.png)
However, if we are to provide the smallest interval that contains all the possible values of
given that
, then it would be:
![\[ (6.48, \infty) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/ph67nt7cmbm3bkb5s9vmdn3gj3qbs1l5fy.png)
This interval indicates that
can be any value greater than 6.48, extending to infinity.