For an integer $n$, the inequality
\[x^2 + nx + 15 < 0\]has no real solutions in $x$. Find the number of different possible values of $n$.

There are 15 different possible values of n:

{-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7}

Step-by-step explanation:

The given inequality is

For to have no solution, the discriminant must be less than zero.

This means that:

This implies that:

The integer values are , {-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7}

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1

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